Let $D$ be a bounded domain in the plane which is symmetric and convex
with respect to both coordinate axes. We prove that the Brownian motion
conditioned to remain forever in $D$, the Doob $h$-process where $h$ is
the ground state Dirichlet eigenfunction in $D$, has the "hot-spots"
property. That is, the first non-constant eigenfunction corresponding
to the semigroup of this process with its nodal line on one of the
coordinate axes attains its maximum and minimum on the boundary and
only on the boundary of the domain. This is the exact analogue for
conditioned Brownian motion of the result in \cite{JN} for Neumann
eigenfunctions.