  We completely characterize the boundedness of the Volterra type integration
operators $J_b$ acting from the weighted Bergman spaces $A^p_\alpha$ to the
Hardy spaces $H^q$ of the unit ball of $\mathbb{C}^n$ for all $0<p,q<\infty$. A
partial solution to the case $n=1$ was previously obtained by Z. Wu in
\cite{Wu}. We solve the cases left open there and extend all the results to the
setting of arbitrary complex dimension $n$. Our tools involve area methods from
harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities,
factorization tricks for tent spaces of sequences, as well as techniques and
integral estimates related to Hardy and Bergman spaces.
