We determine the exact short-time distribution -lnPfH,t=SfH/t of the one-point height H=h(x=0,t) of an evolving 1+1 Kardar-Parisi-Zhang (KPZ) interface for flat initial condition. This is achieved by combining (i) the optimal fluctuation method, (ii) a time-reversal symmetry of the KPZ equation in 1+1 dimension, and (iii) the recently determined exact short-time height distribution -lnPstH,t=SstH/t for stationary initial condition. In studying the large-deviation function SstH of the latter, one encounters two branches: an analytic and a nonanalytic. The analytic branch is nonphysical beyond a critical value of H where a second-order dynamical phase transition occurs. Here we show that, remarkably, it is the analytic branch of SstH which determines the large-deviation function SfH of the flat interface via a simple mapping SfH=2-3/2Sst2H.