The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-"first" and "second" class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.