Step 1: Recall which project the dam belongs to.
Gandhi Sagar Dam is a key structure of the Chambal Valley Project.
Step 2: Eliminate the other rivers.
It has no connection with the Ganges, Kaveri or Mahanadi.
Step 3: Confirm the river.
The dam is built across the Chambal river in Madhya Pradesh.
\[ \boxed{\text{The Chambal}} \]