In this paper we give a conjugate duality approach for a strong bilevel programming problem (S). The approach is based on the use of a regularization of problem (S) and the so-called Fenchel-Lagrange duality. We first show that the regularized problem of (S) admits solutions and any accumulation point of a sequence of regularized solutions solves (S). Then, via this duality approach, we establish necessary and sufficient optimality conditions for the regularized problem. Finally, necessary and sufficient optimality conditions are given for the initial problem (S). We note that such an approach which allows us to apply the Fenchel-Lagrange duality to the class of strong bilevel programming problems is new in the literature. An application to a two-level resource allocation problem is given.