Research Study to Determine if an Experimental Agent, LLME Can Decrease the Incidence and Severity of Graft-Versus-Host-Disease (GVHD) Following Blood (Hematopoietic) Stem Cell Transplantation

Inclusion Criteria: * Patients must be \> 18 years of age, with no upper age limit. * Patients must have an ECOG performance status of 0 or 1. * Any patient with a hematologic malignancy which is unlikely to be cured by conventional treatment is eligible for this study. * Patients for whom a disease specific protocol exists will be transplanted on those protocols as discussed in the introduction. * Patients who have had prior autografts may be treated on this protocol. * Patients must have adequate physical function as measured by the following criteria: * Cardiac: Asymptomatic or, if symptomatic, then left ventricular ejection fraction at rest must be \>40%. * Hepatic: Aspartate transaminase (AST) micro 3x the upper limits of normal and total serum bilirubin \< 2.5 mg/dL. Patients with a higher bilirubin from "benign conditions" such as Gilbert's disease may still be eligible for the study. * Renal: Serum creatinine within the normal range or if creatinine outside normal range then creatinine clearance \> 60 ml/min/1.73m2. Serum creatinine must be less than or equal to 2.0 mg/dl. * Pulmonary: Asymptomatic or, if symptomatic, DLCO (diffusion capacity) \> 45% of predicted (corrected for hemoglobin) * The patient or guardian(s) must be able to give informed consent to the study. * Patient must have a suitable donor who is identical for HLA (human leukocyte antigens) -A, -B, -C, -DR. Single antigen mismatches for HLA-A, -B, -C, -DR are also permitted. Donors obtained through the National Marrow Donor Program (NMDP) will follow NMDP guidelines. Exclusion Criteria: * Patients who are eligible for a standard myeloablative transplant and for whom a standard myeloablative transplant is preferable will not be treated on this protocol.