It is known that the product $\omega_1 \times X$ of $\omega_1$ with an $M_1$-space may be nonnormal. In this paper we prove that the product $\kappa \times X$ of an uncountable regular cardinal $\kappa$ with a paracompact semi-stratifiable space is normal if{f} it is countably paracompact. We also give a sufficient condition under which the product of a normal space with a paracompact space is normal, from which many theorems involving such a product with a countably compact factor can be derived.