Mesothelin-Targeted Immunotoxin LMB-100 in People With Malignant Mesothelioma

* INCLUSION CRITERIA (All Cohorts): * Histologically confirmed epithelial or biphasic 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, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as greater than or equal to 20 mm with conventional techniques or as greater than or equal to10 mm with spiral computed tomography (CT) scan. * 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 curable 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 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 (ECOG) performance status (PS) 0 or1 * Adequate hematological function: neutrophil count of more than of equal to 1.5x10\^9 cells/L, platelet count of greater than or equal to 100,000/microl, (transfusion independent, defined as not receiving platelet transfusions within 7 days prior to laboratory sample) hemoglobin more than or equal to 9 g/dL * Adequate Liver function: aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than 2.5 X upper limit of normal alkaline phosphate less than 2.5 X upper limit of normal unless bone metastasis is present (less than 5 X upper limit of normal) in the absence of liver metastasis. * Bilirubin less than or equal to 1.5 mg/dL (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: * Unconjugated hyperbilirubinemia noted on several occasions * No evidence of hemolysis (normal hemoglobin, reticulocyte count, and lactate dehydrogenase (LDH) * Normal liver function tests * Absence of other diseases associated with unconjugated hyperbilirubinemia * Adequate renal function: creatinine less than 1.5 mg/dL OR creatinine clearance (by Cockcroft Gault formula) greater than or equal to 50 mL/min. * Must have serum albumin \> 2.5 mg/dL without intravenous supplementation * Must have left ventricular ejection fraction \> 50% * Must have an ambulatory oxygen saturation of \> 90% on room air * Women of child-bearing potential (defined as a sexually mature woman who (1) has not undergone hysterectomy \[the surgical removal of the uterus\] or bilateral oophorectomy \[the surgical removal of both ovaries\] or; (2) has not been naturally postmenopausal for at least 24 consecutive months \[i.e., has had menses at any time during the preceding 24 consecutive months\]) must: * Either commit to true abstinence\* from heterosexual contact (which must be reviewed on a monthly basis), or agree to use, and be able to comply with, effective contraception without interruption, 28 days prior to starting study therapy (including dose interruptions), while on study medication and for 3 months after the last dose of study therapy; and * Have a negative serum pregnancy test (\<= -human chorionic gonadotropin (hCG) result at screening and agree to ongoing pregnancy testing during the course of the study, and after the end of study therapy. This applies even if the subject practices true abstinence\* from heterosexual contact. * Men must agree to practice true abstinence or agree to use a condom during sexual contact with a pregnant female or a female of childbearing potential while participating in the study, during dose interruptions and for 6 months following discontinuation of study therapy, even if he has undergone a successful vasectomy. * Ability of subject to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA (All Cohorts): -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, unstable arrhythmias, unstable angina, 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 * 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 * Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with LMB-100, breastfeeding should be discontinued if the mother is treated with LMB-100. These potential risks may also apply to other agents used in this study. * Known hypersensitivity to any of the components of LMB-100 * High doses of systemic corticosteroids within 7 days prior to first dosing. High dose is considered as \> 20 mg of dexamethasone a day (or equivalent) for \> 7 consecutive days. EXCLUSION CRITERIA (Cohort B only) * Subjects must not have received paclitaxel nor nab-paclitaxel within 4 months prior to initiation of study therapy. * Participants with contra-indication and/or history of sever hypersensitivity reactions to nab-paclitaxel.