In patients who demonstrate a complete clinical and hematologic response to imatinib therapy and who do not have life-threatening disease, the dose will be decreased gradually to 100 mg daily and then discontinued. In the event of clinical, hematologic or molecular relapse during the taper, the imatinib dose will be increased to a maximum of 600 mg daily to achieve a second remission. Laboratory monitoring will be performed as above except for molecular monitoring which will be monitored monthly if drug is discontinued or molecular relapse occurs. Once a stable dosing regimen is achieved for greater than or equal to 6 months in subjects who have undergone dose descalation or greater than or equal to 2 years in subjects receiving 300-400 mg of imatinib daily who did not qualify for dose de-escalation, the frequency of NIH visits and end organ assessments will be decreased to 6 months, with molecular monitoring every 3 months and monthly routine laboratory assessments.