Study to Evaluate sUA Lowering Activity, Safety and Efficacy of Oral Ulodesine Added to Allopurinol

Inclusion Criteria: * Age ≥ 18 to \< 70 years * Baseline sUA ≥ 6.0 mg/dL after at least 2 weeks of treatment on a stable 300 mg dose of allopurinol * Documented diagnosis of gout according to the American Rheumatism Association --Preliminary Criteria for the Diagnosis of Gout * Be willing and able to take colchicine 0.6 mg per day or naproxen 220-250 mg twice daily as prophylaxis for gout flares and, if needed, a single daily dose of a proton pump inhibitor to prevent gastrointestinal discomfort * Female participants must: * Be sexually abstinent * Be surgically sterile * Be postmenopausal or on stable contraception * Male participants must: * Be sexually abstinent * Be \> 1 year post-vasectomy * Using condoms with spermicide with partners meeting female requirements Exclusion Criteria: * Unable to tolerate 300 mg allopurinol * Unable to tolerate both colchicine 0.6 mg per day and naproxen 220-250 mg twice daily * Prior participation in a clinical study with BCX4208 * Gout flare during the Screening Period that is resolved for less than 2 weeks prior to first treatment with BCX4208 or placebo (exclusive of chronic synovitis/ arthritis) * Unstable angina, symptomatic arrhythmia, signs or symptoms compatible with New York Heart Association Class III or Class IV heart failure, history of long QT syndrome, or QTc interval \< 350 msec or \> 475 msec * Poorly controlled hypertension (SBP \> 160 mmHg or DBP \> 100 mmHG at Screening or Baseline) * Moderate or severe renal impairment and/or calculated creatinine clearance of \< 60 mL/min(Cockroft-Gault method) * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) values \> 2.0 x ULN * CD4+ cell counts by flow cytometry \< 500 cells/mm3 * Hemoglobin \< 10 g/dL or \> 18 g/dL males or \< 10 g/dL or \> 17 g/dL females White blood cell count \< 3.7 x 109/L or \> 11 x 109/L * Female subjects who are pregnant, breastfeeding, or planning a pregnancy within the next 4 months * Positive serology for hepatitis B surface antigen or hepatitis C or HIV type 1 * Immunocompromised or on systemic immunosuppressive medications or antirheumatic drugs (including anakinra and adrenocorticotropic hormone)from Screening to Day 92 * Azathioprine or 6-mercaptopurine within 14 days of first dose of allopurinol Hydrochlorothiazide in doses \> 50 mg per day from Screening to Day 92 * Use of herbal/dietary supplements Screening to Day 92 * Recipient of any live or attenuated vaccine within 6 weeks of Screening * Uric acid-lowering drugs other than allopurinol from Screening to Day 92 Systemic corticosteroids within 4 weeks prior to Day 1 (this does not include pulmonary or nasal inhaler containing corticosteroids, ophthalmic corticosteroids, joint injections, or low potency topical steroids) * Investigational drug within 30 days prior to signing the ICF for this study * Clinically significant and relevant drug allergies * Chronic or recurrent infections (≥ 3 infections at the same site within 12 months) * Cancer within 12 months-Except non-melanomatous localized skin cancer or completely excised and cured carcinoma-in-situ of uterine cervix or subjects who were previously treated for prostate or breast cancer if they currently are stable and have not been on chemo therapy within the last year prior to screening. * Alcohol or drug abuse within 12 months of signing the ICF, or current substance dependence or abuse * Other medical conditions which, in the opinion of the PI, would jeopardize the safety of the study subject or impact the validity of the study results.