We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions — the crossover distributions — which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.