Abstract Recent research has demonstrated a connection between Weil-etale cohomology and special values of zeta functions. In particular, Lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a Weil-etale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology. We define Weil-etale cohomology for varieties over p-adic fields, and prove a duality theorem for the cohomology of G m on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a p-adic analogue of Lichtenbaum's Weil-etale duality theorem for curves over finite fields, as well as a Weil-etale analogue of his classical duality theorem for curves over p-adic fields. Finally, we show that our duality theorem implies this latter classical duality theorem for index 1 curves.