The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let C n denote the cycle of order n and P 6,6 n the graph obtained from joining two cycles C 6 by a path P n−12 with its two leaves. Let B n denote the class of all bipartite bicyclic graphs but not the graph R a,b , which is obtained from joining two cycles C a and C b (a, b ≥ 10 and a ≡ b ≡ 2 ( mod 4)) by an edge. In [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001), 1002-1005], Gutman and Vidović conjectured that the bicyclic graph with maximal energy is P 6,6 n , for n = 14 and n ≥ 16. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427 (2007), 87-98], Li and Zhang showed that the conjecture is true for graphs in the class B n . However, they could not determine which of the two graphs R a,b and P 6,6 n has the maximal value of energy. In [B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008), 431-433], numerical computations up to a + b = 50 were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of P 6,6 n is larger than that of R a,b , which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.