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.
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