  We study the atypically large deviations of the height $H \sim {{\cal O}}(t)$
at the origin at late times in $1+1$-dimensional growth models belonging to the
Kardar-Parisi-Zhang (KPZ) universality class. We present exact results for the
rate functions for the discrete single step growth model, as well as for the
continuum KPZ equation in a droplet geometry. Based on our exact calculation of
the rate functions we argue that models in the KPZ class undergo a third order
phase transition from a strong coupling to a weak coupling phase, at late
times.
