Inclusion Criteria:  Histologically confirmed dedifferentiated liposarcoma (LPS) or leiomyosarcoma ATX-101 in Sarcoma Phase II (LMS). Pathology review occurs at the center enrolling the patient on this trial. Disease must be locally advanced and unresectable or metastatic. Disease which may be resected but with an associated level of morbidity deemed unacceptable by the treating clinician is considered eligible. Patients must have measurable disease by RECIST criteria version 1.1. In addition, the first 10 patients enrolled on the study must have a site of disease amenable to image-guided biopsy at minimal risk or less, and must agree to undergo this biopsy. Patients must evidence of disease progression, either clinically or radiographically, within the 12 weeks prior to study enrollment, as determined by the investigator enrolling the patient on the study. Patients must have been treated with at least one prior systemic regimen for advanced sarcoma: LMS: Anthracycline-based chemotherapy, or gemcitabine/docetaxel. LPS: No specification as to the prior treatment received. Neoadjuvant or adjuvant systemic therapy does not qualify as prior treatment unless completed within 6 months of disease relapse. Patients must be age 18 years or older. Because the safety of ATX-101 in patients less than 18 years of age has not been characterized, children are excluded from the present study. Patients must demonstrate an Eastern Cooperative Oncology Group (ECOG) performance status of ≤2. Patients must demonstrate normal organ and marrow function as defined below: Absolute neutrophil count (ANC) ≥ 1,500/mm3 Platelet count ≥ 100,000/mm3 Creatinine ≤ 1.5 times upper limit of normal OR Calculated creatinine clearance > 45 mL/min* Total bilirubin ≤ 1.5 times upper limit of normal** Aspartate transaminase (AST)/aminotransferase (ALT) ≤ 1.5 times upper limit of normal** Notes: Upper limit of normal is defined by the clinical laboratory performing the test. Using the lean body mass formula only (Modified Cockcroft Gault) ** If transaminase elevation and/or bilirubin elevation is attributed to the presence of liver metastases, a total bilirubin ≤ 2.5 times the upper limit of normal and an AST and ALT ≤ 2.5 times the upper limit of normal are permissible. Patients with an elevated bilirubin level that is attributed to an inherited disorder, such as Gilbert's disease, may be enrolled at the discretion of the principal investigator. The effects of ATX-101 on the developing human fetus are unknown. For this reason, women of child-bearing potential and all men must agree to use adequate contraception (for women: hormonal or barrier method of birth control, abstinence; for men: male condom, prior vasectomy, or abstinence) prior to study entry and for the duration of study participation. 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. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of ATX-101 administration. If patients do not agree to the above, they are not considered eligible. Ability to understand and willingness to sign a written informed consent document.  Exclusion Criteria  Patients must not have received treatment with any chemotherapy, immunotherapy, radiotherapy or an investigational agent for malignancy within the 21 days of initiating treatment on this protocol. Patients may not have received treatment with a small molecule targeted agent (including off-label or investigational use) within 14 days of initiating treatment on this protocol, provided this represents at least 7 half-lives for that agent. Toxic effects from any prior therapy (except alopecia) must have resolved to grade 1 or less according to NCI CTCAE v4.0 or to the patient's baseline by the time of initiating treatment on this protocol. Patients may not have uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, uncontrolled cardiac arrhythmia, cerebrovascular accident within the last six months, uncontrolled diabetes mellitus, uncontrolled psychiatric illness or any other disease condition that would limit compliance with study requirements in the opinion of the principal investigator. Patients may not be pregnant or nursing. Pregnant women are excluded from this study because the teratogenic effects of ATX-101 have not been adequately studied. A negative pregnancy test must be documented 7 days or less prior to initiating treatment on this protocol. Because there is an unknown but potential risk for adverse events to nursing infants secondary to treatment of the mother with ATX-101, breastfeeding must be discontinued prior to enrollment. Patients may not have known active hepatitis B or C infection. In patients with a history of hepatitis B or C infection, resolution of infection must be demonstrated by negative serology for hepatitis B surface antigen (HBsAg) and/or negative hepatitis C virus (HCV) RNA. Patients may not have uncontrolled HIV/AIDS infection. However, HIV positive patients on highly active retroviral therapy (HAART) with an undetectable viral load and CD4 T-cell count above 200 may participate. Anticipated requirement for surgery during the study period or major surgery within 3 weeks of initiating treatment. Active central nervous system (CNS) metastases or leptomeningeal involvement. Patients with known CNS metastases must have received definitive radiotherapy or surgery at least 4 weeks prior to initiating treatment with imaging demonstrating no progression of disease over this interval.