  For $0<p<\infty $, the Dirichlet-type space $\Dp$ consists of those analytic
functions $f$ in the unit disc $\D$ such that $\int_\D|f'(z)|\sp
p(1-|z|)^{p-1}\,dA(z)<\infty$. Motivated by operator theoretic differences
between the Hardy space $H^p$ and $\Dp$, the integral operator {displaymath}
T_g(f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta,\quad z\in\D, {displaymath}
acting from one of these spaces to another is studied. In particular, it is
shown, on one hand, that $T_g:\Dp\to H^p$ is bounded if and only if $g\in\BMOA$
when $0<p\le 2$, and, on the other hand, that this equivalence is very far from
being true if $p>2$. Those symbols $g$ such that $T_g:\Dp\to H^q$ is bounded
(or compact) when $p<q$ are also characterized. Moreover, the best known
sufficient $L^\infty$-type condition for a positive Borel measure $\mu$ on $\D$
to be a $p$-Carleson measures for $\Dp$, $p>2$, is significantly relaxed, and
the established result is shown to be sharp in a very strong sense.
