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 as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional
             techniques or as ≥ 10 mm with CT scan, MRI, or calipers by clinical exam. See Section
             11 for the evaluation of measurable disease. 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, defined as any one of the following:

               1. A metastatic lesion that is not radioiodine-avid on a diagnostic radioiodine scan
                  performed up to 2 years prior to enrollment in the current study, or

               2. A radioiodine-avid metastatic lesion which remained stable in size or progressed
                  despite radioiodine treatment 6 months or more prior to entry in the study. There
                  are no size limitations for the index lesion used to satisfy this entry
                  criterion.

               3. The presence of at least one fluorodeoxyglucose (FDG) avid lesion with a SUVmax ≥
                  5.

          -  No recent treatment for thyroid cancer as defined as:

               1. No prior 131I therapy is allowed < 6 months prior to initiation of therapy on
                  this protocol. A diagnostic study using < 10 mCi of 131I is not considered 131I
                  therapy.

               2. No external beam radiation therapy < 4 weeks prior to initiation of therapy on
                  this protocol. (Previous treatment with radiation for any indication is allowed
                  if the investigator judges that the previous radiation does not significantly
                  compromise patient safety on this protocol.)

               3. No chemotherapy or targeted therapy (e.g., tyrosine kinase inhibitor) is allowed
                  < 4 weeks prior to the initiation of therapy on this protocol.

          -  Age ≥ 18 years.

          -  ECOG performance status ≤ 2 (or Karnofsky ≥60%

          -  Life expectancy of greater than 3 months.

          -  Able to swallow and retain orally-administered medication and does not have any
             clinically significant gastrointestinal abnormalities that may alter absorption such
             as malabsorption syndrome or major resection of the stomach or bowels.

          -  All prior treatment-related toxicities must be CTCAE v4.0 grade ≤ 1 (except alopecia).
             Grade 2 prior treatment related toxicities may be allowed after discussion with the
             Principal Investigator.

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

          -  Absolute neutrophil count (ANC) ≥ 1.5x109/L

          -  Hemoglobin ≥ 9 g/dL

          -  Platelets ≥ 100 x 109/L

          -  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

          -  Negative pregnancy test within 7 days prior to starting the study premenopausal women.
             Women of non-childbearing potential may be included without pregnancy test if they are
             either surgically sterile or have been postmenopausal for ≥ 1 year.

          -  Fertile men and women must use an effective method of contraception during treatment
             and for at least 6 months after completion of treatment as directed by their
             physician. Effective methods of contraception are defined as those which result in a
             low failure rate (i.e., less than 1% per year) when used consistently and correctly
             (for example implants, injectables, combined oral contraception or intra-uterine
             devices). At the discretion of the Investigator, acceptable methods of contraception
             may include total abstinence in cases where the lifestyle of the patient ensures
             compliance. (Periodic abstinence [e.g., calendar, ovulation, symptothermal,
             postovulation methods] and withdrawal are not acceptable methods of contraception.)

          -  Ability to understand and the willingness to sign a written informed consent document.

          -  Patients must agree to undergo two research biopsies of a malignant lesion. Patients
             may be exempt from biopsy if 1) the investigator or person performing the biopsy
             judges that no tumor is accessible for biopsy, 2) the investigator or person
             performing the biopsy feels that the biopsy poses too great of a risk to the patient,
             or 3) the patient's platelet count is <100,000/mcl or he/she can not be safely removed
             from anti-coagulation therapy (if the anti-coagulation therapy needs to be temporarily
             held for the biopsy procedure). If the only tumor accessible for biopsy is also the
             only lesion that can be used for RECIST v1.1 response evaluation, then the patient may
             be exempt from biopsy. The goal will be to have a minimum of 3 patients undergo one or
             both of these research biopsies. Accrual may be limited only to subjects whose tumor
             is safely accessible for biopsy to ensure the accrual goal for research biopsies
             described above is met (e.g., if 7 of 10 patients are accrued without any biopsies
             having been obtained, then all subsequent subjects who are registered must qualify for
             attempted research biopsy in order to be enrolled.)

          -  Availability of archival tumor tissue from the thyroid cancer primary or metastasis (a
             tissue block or a minimum of 30 unstained slides would be required. Patients with less
             archival tissue available may still be eligible for the study after discussion with
             the MSK Principal Investigator.)

        Exclusion Criteria:

          -  Concomitant malignancies or previous malignancies within the last 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 during this study.

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

          -  Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs
             chemically related to vemurafenib or thyrotropin alpha (Thyrogen).

          -  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

          -  History or evidence of current, clinically significant uncontrolled arrhythmias
             (exception: patients with controlled atrial fibrillation for > 30 days prior to the
             initiation of therapy on this protocol are eligible).

          -  History of acute coronary syndromes (specifically, myocardial infarction and unstable
             angina), severe/unstable angina, coronary angioplasty, or stenting within 6 months
             prior to the initiation of therapy on this protocol.

          -  History of symptomatic congestive heart failure within 6 months prior to the
             initiation of therapy on this protocol.

          -  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.

          -  Pregnant, lactating, or breast feeding women.

          -  Patients unable to follow a low iodine diet or requiring medication with high content
             in iodide (amiodarone).

          -  Patients who received iodinated intravenous contrast as part of a radiographic
             procedure within 3 months of study registration. Those that have had iodinated
             intravenous contrast within this time frame may still be eligible if a urinary iodine
             analysis reveals that the excess iodine has been adequately cleared after the last
             intravenous contrast administration.

          -  Unwillingness or inability to comply with study and follow-up procedures.