  We study the directed polymer of length $t$ in a random potential with fixed
endpoints in dimension 1+1 in the continuum and on the square lattice, by
analytical and numerical methods. The universal regime of high temperature $T$
is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x =
T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with
$\delta$-function disorder correlation. Using the Bethe Ansatz solution for the
attractive boson problem, we obtain all positive integer moments of the
partition function. The lowest cumulants of the free energy are predicted at
small time and found in agreement with numerics. We then obtain the exact
expression at any time for the generating function of the free energy
distribution, in terms of a Fredholm determinant. At large time we find that it
crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$
infinite $t$ limit. The exact free energy distribution is obtained for any time
and compared with very recent results on growth and exclusion models.
