  We consider a planar Brownian loop $B$ that is run for a time $T$ and
conditioned on the event that its range encloses the unusually high area of
$\pi T^2$, with $T$ being large. We study the deviation of the range of the
conditioned process $X$ from a circle of radius $T$, as a model for the
fluctuation of a phase boundary. This deviation is measured by means of the
inradius and outradius of the region enclosed by the range of $X$. We prove
that in a typical realization of the conditioned measure, each of these
quantities differs from $T$ by at most $T^{2/3 + \epsilon}$.
