  The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p
\times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved
under the assumption that their symbols are in the bilinear H\"ormander class
$BS^m_{\rho,\rho}$, $0 \le \rho <1$, of critical order $m$, where $H^p$ is the
Hardy space. This combined with the previous results of the same authors
establishes the sharp boundedness from $H^p \times H^q$ to $L^r$,
$1/p+1/q=1/r$, of those operators in the full range $0< p, q \le \infty$, where
$L^r$ is replaced by $BMO$ if $r=\infty$.
