  We study the elastic (1+1)-dimensional string subject to a random gaussian
potential on scales smaller than the correlation radius of the disorder
potential (Larkin problem). We present an exact calculation of the probability
function ${\cal P} [F(u,L)]$ for the free energy $F$ of a string starting at
$(0,0)$ and ending at $(u,L)$. The function ${\cal P}(F)$ is strongly
asymmetric, with the left tail decaying exponentially ($\ln {\cal
P}(F\to-\infty)\propto F$) and the right tail vanishing as $\ln {\cal P}(F\to
+\infty)\propto -F^{3}$. Our analysis defines a strategy for future attacks on
this class of problems.
