We investigate when the .product of two smooth manifolds admits a weakly Lagrangian embedding. Assume M, N are oriented smooth manifolds of dimension m and n,. respectively, which admit weakly Lagrangian immersions into $C^m$ and $C^n$. If m and n are odd, then $M;{ imes};N$ admits a weakly Lagrangian embedding into $C^{m+n}$ In the case when m is odd and n is even, we assume further that $chi$(N) is an even integer. Then $M;{ imes};N$ admits a weakly Lagrangian embedding into $C^{m+n}$. As a corollary, we obtain the result that $S^n_1;{ imes};S^n_2;{ imes};...{ imes};S^n_k$, $kappa$>1, admits a weakly Lagrang.ian embedding into $C^n_1+^n_2+...+^n_k$ if and only if some ni is odd.