Step 1: Recall what Ogilvie's syndrome actually is. Its alternative name, acute colonic pseudo-obstruction, gives the answer away because the word colonic names the involved organ.
Step 2: The disorder produces all the features of a blocked large bowel, distension, radiographic dilatation, and obstruction signs, yet no physical blockage exists. The bowel simply stops propelling its contents.
Step 3: The large intestine balloons, and the caecum dilates the most. Because the caecum has the widest diameter and thinnest wall, it is at greatest risk of bursting if the colon is not decompressed.
Step 4: Typical patients are immobile hospital inpatients on narcotics with major illness. Putting this together, the organ involved is the colon, not the stomach, gallbladder or small bowel.
\[\boxed{\text{Colon}}\]