Prior to the study, patients will undergo a 1- to 2-week screening period to assess their eligibility for treatment. Eligible patients will participate in the study for up to 16 cycles of treatment, with each cycle lasting approximately 4 weeks. For each cycle, patients will receive 1 prescribed dose of BL22 every other day for a total of 3 doses per cycle, and will be assessed after every cycle to evaluate the success of the treatment. During the evaluation visits, patients will be required to have a brief physical examination, give blood and urine samples for testing, and undergo other tests as need to check heart and kidney function and assess the state of the leukemia. Patients who agree will give additional blood, urine, or bone marrow samples for future research purposes.