This is a Phase I/II, open-label, escalating dose and number of doses study that will involve up to 30-40 patients with B-CLL. A patient who fails to reach their final assessment visit due to reasons other than the occurrence of a dose limiting toxicity (DLT) may be replaced. Eligible patients will be enrolled into Part I or Part II of the study in cohorts of three patients. There will be a requirement to stagger Day 1 dosing between all patients in each cohort by a minimum of 48 hours. In the first part of the study (Part I) patients will receive a single dose of acadesine on Day 1. In the second part of the study (Part II), patients will receive up to 5 doses of treatment over a period of up to 20 days starting on Day 1.