Alzheimer's disease (AD) is a fatal degenerative disease of the brain for which there is no cure. AD causes brain cells to die. AD is thought to be caused by an excess of A-Beta amyloid, a sticky protein in the brain that forms amyloid plaques. At autopsy, AD patients are required to have these amyloid plaques in the brain in order to have a definitive diagnosis of AD. Inhibiting the enzyme gamma-secretase inhibits the production of A-Beta amyloid as measured in blood and spinal fluid in humans tested thus far and in blood, spinal fluid and brain in animals tested thus far. This study will use several different tests to measure the effect of LY450139 on both A-Beta amyloid and amyloid plaques for some patients. The build up of amyloid plaques will be measured by a new brain scan that can take a picture of amyloid plaques in the brain. Other tests will attempt to measure the overall function of the brain and brain size in some patients. In this trial, patients who initially receive placebo (inactive sugar pill) will at a certain point in the study be switched over to active drug, LY450139. In other words, all patients will eventually receive active drug. Each patient's participation will last approximately two years. Patients taking approved AD medications may participate in this study and continue taking these medications during the study. All patients who complete this study will have the option to continue receiving LY450139 by participating in an open label study.