I. Determine the safety and tolerability of VEGF Trap and docetaxel in patients with persistent or recurrent ovarian epithelial, primary peritoneal, or fallopian tube cancer. (Phase I [closed to accrual as of 3/14/2008]) II. Determine the maximum tolerated dose of VEGF Trap in these patients. (Phase I [closed to accrual as of 3/14/2008]) III. Determine the pharmacokinetics of VEGF Trap when administered alone and in combination with docetaxel in these patients. (Phase I [closed to accrual as of 3/14/2008]) IV. Determine the effects of VEGF Trap on tumor perfusion and metabolism in these patients. (Phase I [closed to accrual as of 3/14/2008]) V. Determine the effect of VEGF Trap and docetaxel on circulating endothelial precursors and circulating endothelial cells in these patients. (Phase I [closed to accrual as of 3/14/2008]) VI. Determine the frequency of clinical response (partial response and complete response) in patients treated with this regimen. (Phase II [open to accrual as of 5/9/2008]) VII. Determine the progression-free survival (PFS) and overall survival (OS) of patients treated with this regimen. (Phase II [open to accrual as of 5/9/2008])