Summary: Let \(\omega\) and \(\nu\) be radial weights on the unit disc of the complex plane, and denote \(\sigma =\omega^{p'} \nu^{-\frac{p'}{p}}\) and \(\omega_x =\int_0^1 \, s^x \omega (s)\,ds\) for all \(1\leq x<\infty\). Consider the one-weight inequality \[ \Vert P_{\omega} (f)\Vert_{L^p_{\nu}}\leq C\Vert f\Vert_{L^p_{\nu}},\quad 1<p<\infty, \qquad \qquad (\dagger) \] for the Bergman projection \(P_{\omega}\) induced by \(\omega\). It is shown that the moment condition \[ D_p (\omega, \nu)=\sup_{n\in\mathbb{N}\cup \{0\}} \frac{\left(\nu_{np+1}\right)^\frac{1}{p}\left(\sigma_{np'+1} \right)^\frac{1}{p'}}{\omega_{2n+1}}<\infty \] is necessary for \((\dagger)\) to hold. Further, \(D_p (\omega, \nu) <\infty\) is also sufficient for \((\dagger)\) if \(\nu\) admits the doubling properties \(\sup_{0\leq r<1}\frac{\int_r^1 \nu (s) s\,ds}{\int_{\frac{1+r}{2}}^1 \nu (s)s\,ds}<\infty\) and \(\sup_{0\leq r<1}\frac{\int_r^1 \nu (s)s\,ds}{\int_r^{1-\frac{1-r}{K}} \nu (s)s\,ds}<\infty\) for some \(K>1\). In addition, an analogous result for the one weight inequality \(\Vert P_{\omega} (f)\Vert_{D^p_{\nu, k}} \leq C\Vert f\Vert_{L^p_{\nu}}\), where \[ \Vert f \Vert_{D^p_{\nu, k}}^p =\sum\limits_{j=0}^{k-1} \vert f^{(j)}(0)\vert^p +\int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-\vert z \vert)^{kp} \nu (z)\,dA(z)<\infty, \quad k\in\mathbb{N}, \] is established. The inequality \((\dagger)\) is further studied by using the necessary condition \(D_p (\omega, \nu)<\infty\) in the case of the exponential type weights \(\nu (r)=\exp \left(-\frac{\alpha}{(1-r^l)^{\beta}}\right)\) and \(\omega (r)= \exp \left(-\frac{\widetilde{\alpha}}{(1-r^{\widetilde{l}})^{\widetilde{\beta}}} \right)\), where \(0<\alpha, \, \widetilde{\alpha},\, l,\, \widetilde{l}<\infty\) and \(0<\beta,\, \widetilde{\beta}\leq 1\).