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Consider a path of planar Brownian motion in the unit disc $D$, started at $0$ and conditioned to exit at $1$. Let $U\subseteq D$ be a simply connected domain containing $0$ and agreeing with $D$ near $1$. Let ${\Phi }_U:U\to D$ be a conformal map normalized by ${\Phi }_U(0)=0$ and ${\Phi }_U(1)=1$. ThenThe conditional probability is the ratio of the boundary Poisson kernel densities for exit at $1$ in $U$ and in $D$. The conformal transformation rule for the Poisson kernel contributes the boundary derivative $|{\Phi }_U^{\prime }(1)|$; the normalized interior image is still $0$, so no additional conformal-radius factor appears. Given avoidance of $D\setminus U$, the mapped path has the original conditioned law, with the conformal Brownian clock accounting for its time parametrization.