Compassionate Use Study of Methylnaltrexone

Inclusion Criteria: * Advanced medical illness (that is; terminal illness, such as incurable cancer or end stage aqcuired immunodeficiency syndrome \[AIDS\]) with a life expectancy of one to six months. * On an opioid regimen for the control of pain/discomfort for at least seven days. * Has opioid-induced constipation. * Stable vital signs and systolic blood pressure greater than or equal to (\>=) 85 millimeters of mercury (mmHg), and diastolic blood pressure \>=45 mmHg (Supine or sitting). * Females of childbearing potential must have a negative pregnancy test (serum or urine). * On a laxative regimen (for example, stool softener and SENNA or equivalent) for at least 3 days prior to treatment. Participants who have discontinued laxatives due to intolerability or lack of efficacy are also eligible. Exclusion Criteria: * Participants with known hypersensitivity to methylnaltrexone, naltrexone or naloxone. * Participants who received any investigational new drug (experimental) except for methylnaltrexone in the previous 30 days. * Participants who are constipated with a disease process suggestive of gastrointestinal obstruction, impaction or diagnosed with a current abdominal pathologic process which may represent a non-opioid cause of bowel dysfunction. * Participants who are constipated and have active, clinically significant diverticulitis. * Participants with a surgically acute abdomen. * Participants being treated with opioids for diarrhea, dyspnea, cough, pulmonary edema, or congestive heart failure. * Individuals with a known drug addiction. * Females who are pregnant or nursing.