Bevacizumab Plus Ipilimumab in Patients With Unresectable Stage III or IV Melanoma

* There are two phases to this research study, Induction Phase and Maintenance Phase. * Induction Phase: Participants will receive ipilimumab by an infusion into a vein or central line at weeks 1, 4, 7 and 10 for a total of 4 infusions. Bevacizumab is also given as an infusion into a vein or central line at weeks 1, 4, 7 and 10 along with ipilimumab and then every 3 weeks by itself. During all cycles of study therapy, the participant will have a physical exam on the first day and undergo blood tests at every study visit. At weeks 1, 4, 7, 10 and 12 a urine sample will be obtained for analysis. * Chest, abdomen and pelvic CT scans will be performed at week 12. If the scans at week 12 show that the participants cancer has remained stable or decreased, they will be asked to have repeat CT scans in three months. * Positron Emission Tomography (PET) scans will be done at week 8 and week 16. * Maintenance Phase: If the scans performed at week 12 show the cancer has improved or stayed the same, then the participant will continue to receive bevacizumab every three weeks and undergo a CT scan every 3 months. Also, every 3 months the participant may be eligible to receive additional doses of ipilimumab in addition to the bevacizumab.