Dementia is a chronic, progressive brain disease that may involve a number of symptoms, including memory loss and changes in personality, behavior, judgment, attention span, language and thought. The most common type of dementia is Alzheimer's disease. Over time, patients with Alzheimer's disease may lose ability to perform daily tasks related to personal care (for example bathing, dressing, eating) and may be unable to handle money or travel to familiar places. Several small clinical trials have shown galantamine to be safe and effective in treating the symptoms associated with Alzheimer's disease. Doses studied have ranged from 22.5 - 45 mg/day, with galantamine administered two or three times daily. Additional information is needed to determine the optimal dose regimen for galantamine in the treatment of Alzheimer's' disease. This multicenter, double-blind, placebo-controlled (neither the patient nor the study doctor know the dose of galantamine being given) study evaluates the safety and effectiveness of galantamine in the treatment of patients with Alzheimer's disease using a flexible dose regimen of two doses of galantamine. The flexible dose regimen was chosen in order to allow the physicians to give each patient an optimal dose of galantamine, to simulate what might be done in clinical practice, rather than to require all patients to receive a fixed dose throughout the study. All patients initially receive placebo for a 1-month period and then receive one of two doses of galantamine (beginning with 4 mg twice daily and gradually increasing to 12 or 16 mg twice daily) or placebo for 3 months. The primary measures of effectiveness include the change from baseline to the end of treatment in the ADAS-cog/11 score (Alzheimer's Disease Assessment Scale: sum of 11 cognitive items) and the CIBIC-plus score (Clinician's Interview Based Impression of Change - Plus Caregiver Input). Additional measures of effectiveness assessed at the end of the treatment include the ADAS-cog/13 score (Alzheimer's Disease Assessment Scale: sum of 13 cognitive items), the Disability Assessment for Dementia (DAD) score and the Neuropsychiatric Inventory (NPI)) score. Safety evaluations (incidence of adverse events, quality of sleep (using the Pittsburgh Sleep Scale), electrocardiograms (ECGs), physical examinations, laboratory tests) are performed throughout the study. Blood samples are taken throughout the study to determine the concentration of drug in the blood. Patients who complete this study will have the option to receive open-label galantamine in a separate study. The study hypothesis is that galantamine administered in either dose is effective in the treatment of Alzheimer's disease as compared with placebo and well tolerated, and that using a flexible dose regimen will allow these patients to receive an individualized dose of galantamine that is effective and well tolerated. Galantamine tablets (or placebo), 12 or 16 mg by mouth twice daily for 3 months, beginning with 4 mg twice daily and gradually increasing to 12 or 16 mg twice daily.