This phase I trial studies the side effects and best dose of donor cord blood T-cells after stem cell transplant in treating patients with relapsed hematological malignancies. After umbilical cord blood transplant, stem cells are collected from the donor's cord blood and stored. The stem cells are then returned to the patient to replace the blood-forming cells that were destroyed by treatment. Removing the T cells and treating them in the laboratory before infusing them in the patient may also help boost the patient's immune system.

PRIMARY OBJECTIVES:

I. To evaluate the safety and maximum tolerated dose (MTD) of infusion of ex vivo expanded cord blood T cells (CLI), in cord blood (CB) transplant recipients with relapsed hematological malignancies.

SECONDARY OBJECTIVES:

I. To determine the complete remission rate and overall response as a result of CLI infusion.

II. To determine the effect of CLI infusion on the chimerism. III. To evaluate the incidence rate and grade of acute graft-versus-host disease (GvHD) after CLI infusion.

IV. To determine the disease-free survival, cytopenia rate, relapse incidence after CLI infusion.

OUTLINE: This is a dose-escalation study of ex vivo-expanded T-cells.

Patients undergo ex vivo-expanded umbilical cord blood progenitor cell donor T cell infusion with aldesleukin 11-14 days after T-cell co-stimulation begins.

After completion of study treatment, patients are followed up for 100 days.

Inclusion Criteria:

• Umbilical cord blood (UCB) recipients with underlying hematological malignancies presenting with post-transplant relapse and have available approximately 400 microliter to 1 ml aliquots or CB wash from previous transplant
• UCB recipients with T-cell and/or overall chimerism value of less than 80%, in absence of relapse and have available approximately 400 microliter to 1 ml aliquots or CB wash from previous transplant
• Performance score of at least 80% by Karnofsky or performance status (PS) < 3 (Eastern Cooperative Oncology Group [ECOG]) (age >= 12 years), or Lansky Play-performance scale of at least 60% or greater (age < 12 years)
• Negative beta human chorionic gonadotropin (HCG) or urine test in females of childbearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on the study
• Patient or patient's legal representative, parent(s) or guardian able to sign informed consent

Exclusion Criteria:

• Human immunodeficiency virus (HIV) positive (due to the extreme immunosuppressive nature of allogeneic stem cell transplant)
• Patients with active (untreated) central nervous system (CNS) disease
• Any active GVHD
• Active invasive infections
• Pregnant or breast-feeding