The purpose of this study is to determine the effect of AT1001 given orally to patients with Fabry disease. Patients will visit the clinic 4 weeks prior to dosing to determine their eligibility for the study, and then return for a second visit for baseline and safety assessments, which will include skin, cardiac, and renal biopsies. Patients will receive oral doses of AT1001 for 24 weeks and will visit the clinic 6 times, once every 4 weeks, for evaluation and tests. A skin biopsy will be repeated after 12 weeks, and then a final set of skin, cardiac, and renal biopsies, and functional assessments will be performed at the end of 24 weeks. Patients may be given the opportunity to enter a study extension phase for an additional 24 weeks, which will require two more clinic visits. All study participants will have a final follow up visit 2 weeks after the end of the study.