  The flow resulting from a strong ultrarelativistic shock moving through a
stellar envelope with a polytrope-like density profile has been studied
analytically and numerically at early times while the fluid temperature is
relativistic--that is, just before and just after the shock breaks out of the
star. Such a flow should expand and accelerate as its internal energy is
converted to bulk kinetic energy; at late enough times, the assumption of
relativistic temperatures becomes invalid. Here we present a new self-similar
solution for the post-breakout flow when the accelerating fluid has bulk
kinetic Lorentz factors much larger than unity but is cooling through $p/n$ of
order unity to subrelativistic temperatures. This solution gives a relation
between a fluid element's terminal Lorentz factor and that element's Lorentz
factor just after it is shocked. Our numerical integrations agree well with the
solution. While our solution assumes a planar flow, we show that corrections
due to spherical geometry are important only for extremely fast ejecta
originating in a region very close to the stellar surface. This region grows if
the shock becomes relativistic deeper in the star.
