  We report on the first exact solution of the KPZ equation in one dimension,
with an initial condition which physically corresponds to the motion of a
macroscopically curved height profile. The solution provides a determinantal
formula for the probability distribution function of the height $h(x,t)$ for
all $t>0$. In particular, we show that for large $t$, on the scale $t^{1/3}$,
the statistics is given by the Tracy-Widom distribution, known already from the
theory of GUE random matrices. Our solution confirms that the KPZ equation
describes the interface motion in the regime of weak driving force. Within this
regime the KPZ equation details how the long time asymptotics is approached.
