Inclusion Criteria:

          1. Signed, written IRB-approved informed consent.

          2. Men and women age ≥ 18 years

          3. ECOG Performance status 0 or 1

          4. For patients with mBCC, histologic confirmation of distant BCC metastasis (e.g., lung,
             liver, lymph nodes, or bone), with metastatic disease that is RECIST measurable using
             CT or MRI

             Phase I Expansion:

             If a patient with locally advanced BCC also has a tumor that is not contiguous with
             cutaneous BCC, e.g., regional lymph nodes (if confirmed on biopsy as BCC and RECIST
             measurable), the patients should be considered as having mBCC and should be enrolled
             in the mBCC cohort

          5. For patients with locally advanced BCC, histologically confirmed disease with at least
             one lesion that was 10 mm or more in at least 1 dimension by color photograph that is
             considered to be inoperable or medical contraindication to surgery (see below), in the
             opinion of a Mohs dermatologic surgeon, head and neck surgeon, or plastic surgeon

          6. Acceptable medical contraindications to surgery include:

               1. BCC that has recurred in the same location after two or more surgical procedures
                  and curative resection is deemed unlikely

               2. Anticipated substantial morbidity and/or deformity from surgery (e.g., removal of
                  all or part of a facial structure, such as nose, ear, eyelid, eye; or requirement
                  for limb amputation)

               3. Other conditions considered to be medically contraindicating must be discussed
                  with the Medical Monitor before enrolling the patient.

          7. For all patients, smoothened inhibitor must have been previously administered for
             their locally advanced or metastatic BCC, unless smoothened inhibitor is inappropriate
             (e.g., patient has received a smoothened inhibitor but became intolerant to the
             therapy). For patients whose BCC has been treated with smoothened inhibitor, disease
             must have progressed after treatment.

          8. For patients with locally advanced BCC, radiotherapy must have been previously
             administered for their locally advanced BCC, unless radiotherapy is contraindicated or
             inappropriate (e.g., hypersensitivity to radiation due to genetic syndrome such as
             Gorlin syndrome, limitations because of location of tumor, or cumulative prior
             radiotherapy dose). For patients whose locally advanced BCC has been irradiated,
             disease must have progressed after radiation.

          9. Previous Therapy

               -  Surgery: Previous surgery is permitted provided that a minimum of 28 days (4
                  weeks) have elapsed between any major surgery and date of registration, and that
                  wound healing has occurred.

               -  Cytotoxic Chemotherapy: There is no limit to the number of prior regimens
                  received.

               -  Other Systemic Therapy: Previous treatment with Hh pathway antagonists is not
                  allowed (except for Smoothened inhibitors). There is no limit to the other prior
                  therapies received

             Patients must have recovered (to baseline or ≤ grade 1) from all reversible toxicity
             related to prior chemotherapy or systemic therapy and have adequate washout as
             follows:

             Longest of one of the following:

               -  Two weeks,

               -  5 half-lives for investigational agents,

                    -  For anti-cancer therapies with half-lives > 8 days, a washout period of at
                       least 28 days will be acceptable,

               -  Standard cycle length of standard therapies.

         10. Patients with nevoid BCC syndrome (Gorlin syndrome) may enroll in this study but must
             meet the criteria for locally advanced or metastatic disease listed above.

         11. For patients with locally advanced BCC, willingness to consent to biopsy of tumor(s)
             at baseline and during the study, as mandated by the protocol

         12. Adequate hematopoietic capacity, as defined by the following:

               -  Hemoglobin ≥ 9.0 g/dL and not transfusion dependent

               -  Platelets ≥ 100,000/mm3

               -  Absolute neutrophil count ≥ 1500 cells/mm3

         13. Adequate hepatic function, as defined by the following:

               -  AST and ALT ≤ 2.5 times upper limit of normal (ULN) or ≤ 5 times ULN if liver
                  metastases are present

               -  Total bilirubin ≤ 1.5 x ULN or within 3x the ULN for patients with Gilbert
                  disease

               -  Albumin ≥ 3.0 g/dL

         14. Adequate renal function, as defined by the following:

               -  Renal: calculated creatinine clearance >45 mL/min for patients with abnormal,
                  increased, creatinine levels (Cockcroft-Gault formula).

         15. Women/men of childbearing potential must have agreed to use two effective
             contraceptive methods while on study and for 6 months after the last dose of CX-4945
             (see Appendix D for definition of women of childbearing potential and acceptable and
             unacceptable methods of contraception)

        Exclusion Criteria:

          1. Tumor histology consistent with basosquamous carcinoma (basal cell carcinoma with
             squamous differentiation or metatypical carcinoma).

          2. Pregnant or nursing women. NOTE: Women of child-bearing potential and men must agree
             to use adequate contraception (hormonal or barrier method of birth control; or
             abstinence) prior to study entry and for the duration of study participation. Should a
             man father a child, or a woman become pregnant or suspect she is pregnant while
             participating in this study, he or she should inform the treating physician
             immediately.

          3. Concurrent non-protocol-specified anti-tumor therapy (e.g., chemotherapy, other
             targeted therapy, radiation therapy, or photodynamic therapy)

               -  For patients with multiple cutaneous BCCs at baseline that are not designated by
                  the investigator as target lesions, treatment of these non-target BCCs with
                  surgery may be permitted but must be discussed with the Medical Monitor prior to
                  any surgical procedure.

               -  For patients with locally advanced BCC whose target lesion(s) is/are inoperable
                  at baseline but is/are later deemed potentially operable because of tumor
                  response to CX-4945, surgery with curative intent may be permitted but must be
                  discussed with the Medical Monitor prior to any surgical procedure.

          4. History of other malignancies within 3 years of Day 1, except for tumors with a
             negligible risk for metastasis or death, such as adequately treated squamous-cell
             carcinoma of the skin, ductal carcinoma in situ of the breast, or carcinoma in situ of
             the cervix

          5. Active or uncontrolled infections or with serious illnesses or medical conditions
             which would not permit the patient to be managed according to the protocol.

          6. History of other disease, metabolic dysfunction, physical examination finding, or
             clinical laboratory finding giving reasonable suspicion of a disease or condition that
             contraindicates use of an investigational drug or that might affect interpretation of
             the results of the study or renders the patient at high risk from treatment
             complications

          7. Difficulty with swallowing oral medications

          8. Chronic diarrhea (excess of 2-3 stools/day above normal frequency)