This is a clinical trial to be conducted at multiple sites in the USA. Patients diagnosed with moderate to severe Restless Legs Syndrome will be randomly allocated to one of 5 treatment arms in the study. The 5 arms include 4 arms with different doses of aplindore MR tablets and 1 placebo arm. The treatment will be taken once a day. The study is blinded and neither patients, nor the investigators, will know what treatment the patient is receiving. Patients will be assigned a dose and will be maintained at that dose for several weeks (2 treatment arms include a short titration period). The entire study will take about 6 weeks. The study will measure how effective aplindore is in decreasing symptoms of Restless Legs Syndrome, and will also assess the safety and tolerability of aplindore.