Summary: For bipartite graphs \(G_1, G_2,\ldots,G_k\), the bipartite Ramsey number \(b(G_1, G_2,\ldots, G_k)\) is the least positive integer \(b,\) so that any coloring of the edges of \(K_{b,b}\) with \(k\) colors, will result in a copy of \(G_i\) in the \(i\)th color, for some \(i\). We determine all pairs of positive integers \(r\) and \(t\), such that for a sufficiently large positive integer \(s\), any 2-coloring of \(K_{r,t}\) has a monochromatic copy of \(C_{2s}\). Let \(a\) and \(b\) be positive integers with \(a\geq b\). For bipartite graphs \(G_1\) and \(G_2\), the bipartite Ramsey number pair \((a,b)\), denoted by \(b_p(G_1,G_2)=(a,b)\), is an ordered pair of integers such that for any blue-red coloring of the edges of \(K_{r,t}\), with \(r\geq t\), either a blue copy of \(G_1\) exists or a red copy of \(G_2\) exists if and only if \(r\geq a\) and \(t\geq b\). \textit{R. J. Faudree} and \textit{R. H. Schelp} [J. Comb. Theory, Ser. B 19, 161--173 (1975; Zbl 0305.05108)] showed that \(b_p(P_{2s},P_{2s})=(2s-1,2s-1)\), for \(s\geq 1\). In this paper we will show that for a sufficiently large positive integer \(s\), any 2-coloring of \(K_{2s,2s-1}\) has a monochromatic \(C_{2s}\). This will imply that \(b_p(C_{2s}, C_{2s})=(2s,2s-1)\), if \(s\) is sufficiently large.