  We consider the solution of the stochastic heat equation \partial_T
\mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with
delta function initial condition \mathcal{Z} (T=0)= \delta_0 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 obtain explicit formulas for the one-dimensional
marginal distributions -- the {\it 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.
