For \(\alpha>0\), let \(B^\alpha\) be the \(\alpha\)-Bloch class, that is, the space of all holomorphic functions \(f\) in the unit disk \(\mathbb D\) for which \(\| f\|_\alpha:=\sup_{z\in\mathbb D}(1-|z|^2)^\alpha |f '(z)|<\infty\). For positive continuous functions \(\phi\) on \([0,1[\) satisfying \[ \text{\(\frac{\phi(r)}{(1-r)^s} \downarrow 0\), respectively \(\frac{\phi(r)}{(1-r)^t} \uparrow \infty\)} \] for some \(0<s<t\), the author considers the space \(E=H(p,p,\phi)\) of holomorphic functions on \(\mathbb D\) for which \(\| f\|^p_E=\int_{\mathbb D} |f(z)|^p \frac{\phi(|z|)^p}{1-|z|}\,dA(z)<\infty\). A complete description of those functions \(g\) holomorphic on \(\mathbb D\) is given for which the integral operators \(J_gf(z)=\int_0^zf(\xi)g'(\xi)\,d\xi\) and \(I_gf(z)=\int_0^zf'(\xi)g(\xi)\,d\xi\) are bounded, respectively compact, when regarded as maps between \(E\) and \(B^\alpha\) (or vice versa).