Let \(E\) be a Banach space and let \(A\) and \(B\) be bounded operators on \(E\). This paper studies the problem of the weak compactness of the linear multiplication operator \(A\wedge B: S\mapsto BSA\) from \(L(E)\) into \(L(E)\) (\(A,B\neq 0\)). A necessary condition for the weak compactness of \(A\wedge B\) is that \(A\) and \(B\) are weakly compact. In the case when \(E\) is a \({\mathcal L}^ 1\)- or a \({\mathcal L}^ \infty\)-space this is also sufficient. In general, compactness of \(A\) or \(B\) implies that \(A\wedge B\) is weakly compact. Moreover, the precise conditions on \(A\) and \(B\) are determined on several concrete spaces. It turns out that the answer to the problem correlates with several properties of the underlying space \(E\), such as reflexivity, the Radon-Nikodym property, the Dunford-Pettis property and the structure of ideals in \(L(E)\). Further results were subsequently obtained by \textit{G. Racher} [``On the tensor product of weakly compact operators'', Math. Ann. 294, 267-275 (1992)].