We study the short-time behavior of the probability distribution P(H,t) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension. The process starts from a stationary interface: h(x,t=0) is given by a realization of two-sided Brownian motion constrained by h(0,0)=0. We find a singularity of the large deviation function of H at a critical value H=Hc. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x↔-x of optimal paths h(x,t) predicted by the weak-noise theory of the KPZ equation. At |H| |Hc| the corresponding tail of P(H) scales as -lnP∼|H|3/2/t1/2 and agrees, at any t>0, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of P scales as -lnP∼|H|5/2/t1/2 and coincides with the corresponding tail for the sharp-wedge initial condition.