Study of the Combination of DKN-01 and Nivolumab in Previously Treated Patients With Advanced Biliary Tract Cancer (BTC)

Inclusion Criteria: * Histologically confirmed intra- or extrahepatic cholangiocarcinoma or gallbladder cancer * Participants must have measurable disease by CT/MRI by RECIST version 1.1 criteria * Prior chemoembolization, radiofrequency ablation, or radiation to the liver is allowed as long as the patient has measurable disease outside of the treated area or measurable progression per RECIST v1.1 at the site of the treated area. * Documented progression after ≥1 line of systemic therapy for advanced BTC. Prior adjuvant chemotherapy qualifies as this 1 line if the last cycle of adjuvant therapy was completed within 6 months of radiological progression. * Age ≥ 18 years * ECOG performance status ≤1 * Life expectancy of greater than 3 months * Participants must have normal organ and marrow function as defined below: * Absolute neutrophil count ≥1,500/mcL * Absolute lymphocyte count ≥1.0 x 10\^9/L * Platelets ≥75,000/mcL * Hemoglobin ≥ 8.0 g/dL (prior transfusions are allowed if given ≥ 7 days before testing) * Total bilirubin \< 2.0 x institutional upper limit of normal; except patients with Gilbert Syndrome who must have a total bilirubin level of \< 3.0 x ULN * AST(SGOT)/ALT(SGPT) ≤3 × institutional upper limit of normal; \< 5 x ULN in case of liver metastases * Creatinine \< 2.0 x institutional upper limit of normal OR Creatinine clearance ≥30 mL/min/1.73 m2 for participants with creatinine levels ≥ ULN * International Normalized Ratio (INR) ≤ 1.5 x ULN unless patient is receiving anticoagulant therapy as long as INR is within therapeutic range of intended use of anticoagulants * Serum albumin \> 2.5 g/dL * Subjects with hepatitis B or C are eligible to enroll if they have: * Chronic HBV infection (evidenced by a positive HBV surface antigen or HBV DNA) as long as they have been on antiviral therapy for ≥ 4 weeks. * Chronic or resolved HCV infection (evidenced by a detectable HCV RNA or antibody). Antiviral therapy is not required for chronic HCV. * Women of child-bearing potential and men must agree to use adequate contraception according to national guidelines (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 5 months for women and 7 months for men after completion of study drug administration. * Female subjects must be either of non-reproductive potential (i.e., post-menopausal by history: ≥ 50 years old and no menses for ≥1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Prior DKK1 inhibitor or anti-PD-1/PD-L1 treatment * Participants with Child-Pugh B or C cirrhosis * Participants with a diagnosis of ampullary cancer * Treatment with any of the following within the specified time frame prior to the first dose of DKN-01 and nivolumab: * Any non-investigational or investigational anticancer therapy within 3 weeks or have not recovered from side effects of such therapy prior to treatment administration (mitomycin within prior 5 weeks). For targeted therapy, 5 half-lives are sufficient, even if \<3 weeks. Concurrent participation in an observational study may be allowed after review by the Principal Investigator. * Patients with locoregional therapy, e.g., transarterial chemoembolization (TACE), selective internal radiotherapy (SIRT), external beam radiation, or ablation within 4 weeks * Palliative limited field radiotherapy (i.e. bone metastases) within 2 weeks * Major surgery within the previous 4 weeks (the surgical incision should be fully healed prior to the first dose of treatment) * Fredericia's corrected QT interval (QTcF) ≥ 500 ms on ECG conducted during screening * History of allergic reactions attributed to compounds of similar chemical or biologic composition to DKN-01 or Nivolumab. * A serious illness or medical condition(s) including, but not limited to, the following: * Known brain metastasis (not including primary brain tumors) unless patient is clinically stable for ≥ 1 month without systemic corticosteroids beyond physiologic replacement (\>10 mg prednisone daily). * Known acute systemic infection. * Myocardial infarction, severe/unstable angina, symptomatic congestive heart failure * New York Heart Association \[NYHA\] Class III or IV (see Appendix D, New York Heart Association \[NYHA\] Classification) within the previous 2 months; if \>2 months, cardiac function must be within normal limits and the patient must be free of cardiac-related symptoms. * Chronic nausea, vomiting, or diarrhea considered to be clinically significant in the opinion of the investigator. * Congenital long QT syndrome, or any known history of torsade de pointes, or family history of unexplained sudden death. * Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that in the judgment of the investigator would make the patient inappropriate for entry into this study. * Patients with a history of another primary malignancy that is currently clinically significant, and has potential for metastases or currently requires active intervention (except for hormonal therapy for breast or prostate cancer). * Any condition requiring systemic treatment with either corticosteroids (\> 2mg daily dexamethasone equivalent) or other immunosuppressive medications within 14 days of starting the study medications. Premedication for hypersensitivity reactions (e.g. to contrast for CT or gadolinium for MRI) is allowed. * Subjects with autoimmune disease active within the last two years including but not limited to Crohn's disease, ulcerative colitis, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjögren's syndrome, Guillain-Barré syndrome, multiple sclerosis, type I diabetes mellitus, vasculitis, or glomerulonephritis * Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone are eligible for this study. * Patients with controlled Type I diabetes mellitus on a stable insulin regimen are eligible * History of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), idiopathic pneumonitis, or evidence of active pneumonitis on screening chest CT scan. History of radiation pneumonitis/fibrosis in the radiation field is permitted. * Patients who received treatment with live vaccines within 30 days prior to the first dose of study medication. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, chicken pox, shingles, yellow fever, seasonal flu, H1N1 flu, rabies, BCG and typhoid vaccine. * History of osteonecrosis of the hip or evidence of structural bone abnormalities in the proximal femur on magnetic resonance imaging (MRI) scan that is symptomatic and clinically significant. Degenerative changes of the hip joint are not excluded. * Known osteoblastic bony metastasis. Screening of asymptomatic subjects without a history of metastatic bony lesions is not required. * Prior allogeneic stem cell or solid organ transplant. * Known or current evidence of HIV * Pregnant or lactating female.