Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed thyroid carcinoma of
             follicular origin (including papillary, follicular, or poorly differentiated subtypes
             and their respective variants).

          -  Confirmation in a CLIA certified laboratory or in an FDA-approved assay that one of
             the patient's thyroid tumors (primary tumor, recurrent tumor, or metastasis) possesses
             a BRAF mutation at V600.

          -  Patients must have measurable disease defined by RECIST criteria 1.1.

          -  Tumors in previously irradiated fields may be considered measureable if there is
             evidence of tumor progression after radiation treatment.

          -  RAI-refractory disease on structural imaging

          -  Age ≥ 18 years.

          -  ECOG performance status ≤ 2

          -  Patients must have normal organ and marrow function as defined below:

               -  Absolute neutrophil count (ANC) > 1500/mcl

               -  Hemoglobin ≥ 9 g/dL

               -  Platelets ≥ 100,000/mcl

               -  Albumin ≥ 2.5 g/dL

               -  Total bilirubin ≤ 1.5x institutional ULN

               -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2x
                  institutional ULN unless it is related to the primary disease

               -  Creatinine ≤ 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault
                  formula) ≥ 50 mL/min OR 24-hour urine creatinine clearance ≥ 50 mL/min

        Exclusion Criteria:

          -  Concomitant malignancies or previous malignancies treated within the past 3 years.
             Exception: Patients who have been disease-free for 3 years, patients with a history of
             completely resected non-melanoma skin cancer, and/or patients with indolent secondary
             malignancies, are eligible.

          -  Use of other investigational drugs within 28 days preceding the first dose of
             vemurafenib on this study.

          -  Symptomatic or untreated leptomeningeal or brain metastases or spinal cord
             compression.

          -  History or evidence of cardiovascular risk including any of the following:

               -  Corrected QT (QTc) interval ≥ 450 msec at baseline or history of congenital long
                  QT syndrome or uncorrectable electrolyte abnormalities. (Patients with well
                  controlled atrial fibrillation are exempt from this criteria.)

               -  History of cerebrovascular attack or transient ischemic attack within 6 months
                  prior to the initiation of therapy on this protocol.

               -  Uncontrolled intercurrent illness including, but not limited to, ongoing or
                  active infection or psychiatric illness/social situations that would limit
                  compliance with study requirements.