Study participants are randomly assigned to one of two treatment groups. One group receives leuprolide or goserelin followed by thalidomide; the other receives leuprolide or goserelin followed by placebo (a look-alike pill with no active ingredients). Patients in both groups receive an injection of leuprolide or goserelin once a month for 6 months. After that time they take four capsules of either thalidomide or placebo once a day and remain on the drug until their prostate-specific antigen (PSA) level returns to what it was before beginning leuprolide or goserelin or to 5 nanograms per liter, whichever is lower.(PSA is a protein secreted by the prostate gland. Monitoring changes in levels of this protein can help evaluate tumor progression). At this point the entire procedure begins again, starting with leuprolide or goserelin treatment, but the experimental drug is switched; patients originally treated with thalidomide are crossed over to placebo, and patients originally treated with placebo are crossed over to thalidomide.