Mesothelin-Targeted Immunotoxin LMB-100 in Combination With SEL-110 in Subjects With Malignant Pleural or Peritoneal Mesothelioma

* INCLUSION CRITERIA: * Histologically confirmed epithelial or biphasic pleural or peritoneal mesothelioma not amenable to potentially curative surgical resection. However, patients with biphasic tumors that have a more than or equal to 50% sarcomatoid component will be excluded. The diagnosis will be confirmed by the Laboratory of Pathology, Center for Cancer Research (CCR), National Cancer Institute (NCI). * Archival sample or fresh biopsy or tumor effusion must be available for confirmation of diagnosis. * Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. * Patients must have had at least one prior chemotherapy regimen that includes pemetrexed and cisplatin or carboplatin. There is no limit to the number of prior chemotherapy regimens received. * The last dose of previous therapy must have occurred at least 3 weeks prior to the start of study therapy. Palliative radiotherapy is allowed up to 2 weeks before the first LMB-100 infusion. * Patients for whom no standard curative therapy exists * Age greater than or equal to 18 years. Because no dosing or adverse event data are currently available on the use of LMB-100 + SEL-110 in patients \<18 years of age, children are excluded from this study * All acute toxic effects of any prior radiotherapy, chemotherapy, or surgical procedure must have resolved to Grade less than or equal to 1, except alopecia (any grade) and Grade 2 peripheral neuropathy. * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Adequate hematological function: neutrophil count of more than or equal to 1.0 times 10(9) cells/L, platelet count of greater than or equal to 100,000/mcL, hemoglobin more than or equal to 9 g/dL * Adequate liver function: Bilirubin less than or equal to 2.5 times the upper limit of normal (ULN) (excluding Gilbert's Syndrome, see below). * Patients with Gilbert's syndrome will be eligible for the study. The diagnosis of Gilbert's syndrome is suspected in people who have persistent, slightly elevated levels of unconjugated bilirubin without any other apparent cause. A diagnosis of Gilbert's syndrome will be based on the exclusion of other diseases based on the following criteria: 1. Unconjugated hyperbilirubinemia noted on several occasions 2. No evidence of hemolysis (normal hemoglobin, reticulocyte count, and lactate dehydrogenase (LDH)) 3. Normal liver function tests 4. Absence of other diseases associated with unconjugated hyperbilirubinemia * Adequate renal function: creatinine clearance (by Cockcroft Gault formula) greater than or equal to 50 mL/min. * Must have serum albumin \> 2.5 g/dL without intravenous supplementation * Must have left ventricular ejection fraction \> 50% * Must have an ambulatory oxygen saturation of \> 90% on room air * The effects of LMB-100 in combination with SEL-110 on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry until 3 months after the last dose of study therapy. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Ability of subject to understand and the willingness to sign a written informed consent document EXCLUSION CRITERIA: * Known or clinically suspected central nervous system (CNS) primary tumors or metastases including leptomeningeal metastases. History or clinical evidence of CNS metastases unless they have been previously treated, are asymptomatic, and have had no requirement for steroids or enzyme-inducing anticonvulsants in the last 14 days. * Evidence of significant, uncontrolled concomitant diseases which could affect compliance with the protocol or interpretation of results, including significant pulmonary disease other than primary cancer, uncontrolled diabetes mellitus, and/or significant cardiovascular disease (such as New York Heart Association Class III or IV cardiac disease, myocardial infarction within the last 6 months, uncontrolled arrhythmias, unstable angina, non-compensative congestive heart failure, or clinically significant pericardial effusion) * Active or uncontrolled infections. * Human immunodeficiency virus (HIV) or active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection. HIV positive patients will be excluded due to a theoretical concern that the degree of immune suppression associated with the treatment may result in progression of HIV infection. * Patients with prior pneumonectomy * Prior therapy with LMB-100 * Any other diseases, metabolic dysfunction, physical examination finding, or clinical laboratory finding giving reasonable suspicion of a disease or condition that would contraindicate the use of an investigational drug * Major surgery or significant traumatic injury greater than or equal to 28 days prior to the first LMB-100 infusion (excluding biopsies) or anticipation of the need for major surgery during study treatment * Dementia or altered mental status that would prohibit informed consent * Live attenuated vaccinations 14 days prior to treatment * Pregnant women are excluded from this study because it is unknown whether LMB-100 + SEL 100 has the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with LMB-100+SEL-110, breastfeeding should be discontinued if the mother is treated with LMB-100+SEL-110. These potential risks may also apply to other agents used in this study. * Known hypersensitivity to any of the components of LMB-100 and/or SEL-110 * Presence of immunosuppressive conditions, including administration of any medications or treatments that may adversely affect the immune system such as allergy injections, immune globulin, interferon, immunomodulators, cytotoxic drugs, or systemic corticosteroids (oral or injectable) during 3 months prior to enrollment. Inhaled and topical corticosteroids allowed. * Known allergy to polyethylene glycol (PEG)ylated products. * History of anaphylactic reaction to a recombinant protein or hypersensitivity to PEG. * Taken an investigational drug within 4 weeks prior to study drug administration or plans to take an investigational agent during the study. * Taken a strong inhibitor or inducer of cytochrome P450 3A4 (CYP3A4) within 14 days prior to enrollment (see Flockhart Table or similarly updated source for a list of such agents) * Taken drugs known to interact with Rapamune such as cyclosporine, diltiazem, erythromycin, ketoconazole (and other antifungals), nicardipine (and other calcium channel blockers), rifampin, verapamil within 14 days prior to enrollment * Uncontrolled hypertension (above 150/95 mm Hg). * History of end-stage renal disease requiring dialysis. * Serum phosphorus less than 2.0 mg/dL * Organ transplant recipient