To establish the safety and efficacy of two dose levels of ex-vivo cultured adult human mesenchymal stem cells (hMSCs) (Prochymal®) in participants experiencing acute GVHD, Grades II-IV, post hematopoietic stem cells (HSC) transplant.

Protocol 260 - Participants will be randomized with equal probability to the treatment arms (2 million cells/kilogram (kg) of Prochymal® or 8 million cells/kg of Prochymal®) using a stratified block design. The stratification factor is acute GVHD grade. For the purpose of stratification, the GVHD grades are II and III-IV. Treatment with investigational agent will be administered on study Days 1 and 4. Participants will be followed for safety and efficacy until Day 28 after initiation of treatment with the investigational agent, or until withdrawal or death, whichever occurs first.

Protocol 261- This study is designed as a long-term safety follow-up of participants who take part in the preceding clinical study of Prochymal® (Protocol 260) for the treatment of acute GVHD. Participants will be enrolled in Study 261 upon completion of the preceding Study 260.

Protocol 260 Inclusion Criteria:

• Participant must be 18 to 70 years of age inclusive.

• If female and of child-bearing age, participant must be non-pregnant, not breast feeding, and use adequate contraception. Males must use adequate contraception.

• Participant must have newly diagnosed, Grade II-IV acute GVHD requiring therapy. Biopsy for confirmation of GVHD is not mandatory, but is recommended when feasible. Enrollment should not be delayed awaiting biopsy results.

• Participant must have received either full or reduced intensity myeloablative regimens followed by an allogeneic hematopoietic stem cell transplant using bone marrow, peripheral blood stem cell, or cord blood, including donor lymphocyte infusion (DLI).

• Participant must have minimal renal and hepatic function as defined by:

  * Calculated creatinine clearance (CLcr) of > 30 mL/min using the Cockroft-Gault equation.

• Participant must be available for all specified assessments at the study site through study Day 28.

• Participant must provide written informed consent and authorization for use and disclosure of protected health information (PHI).

Protocol 260 Exclusion Criteria:

• Participant has received previous treatment for Grade II-IV acute GVHD (except as noted in Criterion 2).
• Participant has been treated for GVHD with methylprednisolone, > 2mg/kg/day, for more than 72 hours prior to receiving Prochymal®.
• Participant has uncontrolled alcohol or substance abuse within 6 months of randomization.
• Participant has received an investigational agent (not approved by food and drug administration (FDA) for marketed use in any indication) within 30 days of randomization. Participant may not receive an investigational agent during the 28-day study period.
• Participant has any underlying or current medical or psychiatric condition that, in the opinion of the Investigator, would interfere with the evaluation of the participant (e.g., uncontrolled infection, right heart failure, pulmonary hypertension, etc.).
• Participant has unstable arrhythmia.
• Participant is unwilling to sign consent form for the long-term follow-up study, Protocol 261.
• Participant has a known allergy to bovine or porcine products.
• Participant had received transplant for a solid tumor disease.