Numerical simulations for the merger of binary neutron stars are performed in full general relativity incorporating a finite-temperature (Shen's) equation of state (EOS) and neutrino cooling for the first time. It is found that for this stiff EOS, a hypermassive neutron star (HMNS) with a long lifetime (≫10 ms) is the outcome for the total mass ≲3.0M(⊙). It is shown that the typical total neutrino luminosity of the HMNS is ∼3-8×10(53) erg/s and the effective amplitude of gravitational waves from the HMNS is 4-6×10(-22) at f=2.1-2.5 kHz for a source distance of 100 Mpc. We also present the neutrino luminosity curve when a black hole is formed for the first time.