We search for annihilation decay modes of neutral b mesons into pairs of charmless charged hadrons with the upgraded Collider Detector at the Fermilab Tevatron. Using a data sample corresponding to 6 fb(-1) of integrated luminosity, we obtain the first evidence for the B(s)(0)→π(+)π(-) decay, with a significance of 3.7σ, and a measured branching ratio B(B(s)(0)→π(+)π(-))=(0.57±0.15(stat)±0.10(syst))×10(-6). A search for the B(0)→K(+)K(-) mode in the same sample yields a significance of 2.0σ, and a central value estimate B(B(0)→K(+)K(-))=(0.23±0.10(stat)±0.10(syst))×10(-6).