Vemurafenib With Lymphodepletion Plus Adoptive Cell Transfer & High Dose IL-2 Metastatic Melanoma

In this study, these special immune T-cells will be taken from a sample of the participant's tumor tissue that will be surgically removed. Certain parts of these cells will be multiplied, or grown, in the laboratory. They will then be given back to the patient by an infusion in their veins. These cells are called tumor infiltrating lymphocytes (TIL). The investigators want to study the benefits and side effects of TIL when they are given with the following combination of drugs: * Vemurafenib - a type of drug used to slow the growth of certain types of cancer cells. This drug will be given for about three weeks while T-cells are being grown in the lab and then again after T-cell infusion for up to two years. * Fludarabine and cyclophosphamide - two types of chemotherapy drugs. These drugs will be used for what is called lymphodepletion. The purpose of lymphodepletion in this study is to temporarily reduce the number of normal lymphocytes circulating in the patient's body before they are given the T-cells that were grown in the lab. This is so that there will be more "space" for the lymphocytes (T-cells) that will be infused in their veins. * Interleukin-2 (IL-2) - a drug used to help the body's response to treatment on the immune system. A high dose regimen of IL-2 will be given after they receive the infusion of the T-cells. The use of TIL is investigational, meaning it has not been approved by the U.S. Food and Drug Administration (FDA). Vemurafenib and IL-2 have been approved by the FDA for the treatment of metastatic melanoma and melanoma that cannot be surgically removed. The chemotherapy drugs fludarabine and cyclophosphamide, used for lymphodepletion, have been approved by the FDA, but not for the treatment of metastatic melanoma. The combination of vemurafenib followed by lymphodepletion with chemotherapy, TIL infusion, and high dose IL-2 is investigational, and has not been proven to help treat melanoma. This combination is not FDA approved; however, the FDA is allowing its use in this study.