  We study the asymptotic behavior of the Bergman orthogonal polynomials
$(p_n)_{n=0}^{\infty}$ for a class of bounded simply connected domains $D$. The
class is defined by the requirement that conformal maps $\varphi$ of $D$ onto
the unit disk extend analytically across the boundary $L$ of $D$, and that
$\varphi'$ has a finite number of zeros $z_1,\ldots, z_q$ on $L$. The boundary
$L$ is then piecewise analytic with corners at the zeros of $\varphi'$. A
result of Stylianopoulos implies that a Carleman-type strong asymptotic formula
for $p_n$ holds on the exterior domain $\mathbb{C}\setminus\overline{D}$. We
prove that the same formula remains valid across $L\setminus\{z_1,\ldots,z_q\}$
and on a maximal open subset of $D$. As a consequence, the only boundary points
that attract zeros of $p_n$ are the corners. This is in stark contrast to the
case when $\varphi$ fails to admit an analytic extension past $L$, since when
this happens the zero counting measure of $p_n$ is known to approach the
equilibrium measure for $L$ along suitable subsequences.
