  Using the weak-noise theory, we evaluate the probability distribution
$\mathcal{P}(H,t)$ of large deviations of height $H$ of the evolving surface
height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when
starting from a flat interface. We also determine the optimal history of the
interface, conditioned on reaching the height $H$ at time $t$. We argue that
the tails of $\mathcal{P}$ behave, at arbitrary time $t>0$, and in a proper
moving frame, as $-\ln \mathcal{P}\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The
$3/2$ tail coincides with the asymptotic of the Gaussian orthogonal ensemble
Tracy-Widom distribution, previously observed at long times.
