We present a strongly polynomial algorithm to solve integer programs of the form max{cT x: Ax≤ b, xεℤn }, for AεℤmXn with rank(A)=n, bε≤m, cε≤n, and where all determinants of (nXn)-sub-matrices of A are bounded by 2 in absolute value. In particular, this implies that integer programs max{cT x : Q x≤ b, xεℤ≥0n}, where Qε ℤmXn has the property that all subdeterminants are bounded by 2 in absolute value, can be solved in strongly polynomial time. We thus obtain an extension of the well-known result that integer programs with constraint matrices that are totally unimodular are solvable in strongly polynomial time.