This paper develops a new generalized-branching technique ca lled "explicit- constraint branching" (ECB) to improve the performance of branch-and- bound algorithms for solving mixed-integer programs (MIPs). ECB adds structure to a MIP, in the form of auxiliary constraints and auxiliary integer variables, to allow branching on groups of (original) integer variables that would not otherwise be possible. Computational tests on three sets of real- world MIPs demonstrate that ECB often improves solution times over standard branch and bound, sometimes dramatically.