  We show that every bordered Riemann surface, $M$, with smooth boundary $bM$
admits a proper holomorphic map $M\to \Omega$ into any bounded strongly
pseudoconvex domain $\Omega$ in $\mathbb C^n$, $n>1$, extending to a smooth map
$f:\overline M\to\overline \Omega$ which can be chosen an immersion if $n\ge 3$
and an embedding if $n\ge 4$. Furthermore, $f$ can be chosen to approximate a
given holomorphic map $\overline M\to \Omega$ on compacts in $M$ and
interpolate it at finitely many given points in $M$.
