The author gives necessary and sufficient conditions on a weight function \(v\) in order that the Riemann-Liouville operator \(R_\alpha f(x)=\int_0^x f(t)(x-t)^{\alpha-1}dt\) be bounded from \(L^p\) to \(L^q(v)\) when \(1<p,q<\infty\), \(\alpha\in(1/p,1)\) or \(\alpha>1\), and also that the Weyl operator \(W_\alpha f(x)= \int_x^\infty f(t)(t-x)^{\alpha-1}dt\) be bounded from \(L^p(v)\) into \(L^q\) when \((q-1)/q<\alpha<1\) or \(\alpha>1\). The results bring some new insight to the problem of characterizing a two-weight inequality for the operators \(R_\alpha\) and \(W_\alpha\). The author is apparently unaware of the earlier paper by M. Lorente [Can. J. Math. 49, No. 5, 1010-1033 (1997; Zbl 0915.26002)], where the two-weight inequality for \(W_\alpha\) is characterized, but by a condition which involves the operator itself.