Among all of the freezing transitions, that of water into ice is probably the most relevant to biology, physics, geology, or atmospheric science. In this work, we investigate homogeneous ice nucleation by means of computer simulations. We evaluate the size of the critical cluster and the nucleation rate for temperatures ranging between 15 and 35 K below melting. We use the TIP4P/2005 and the TIP4P/ice water models. Both give similar results when compared at the same temperature difference with the model's melting temperature. The size of the critical cluster varies from ∼8000 molecules (radius = 4 nm) at 15 K below melting to ∼600 molecules (radius = 1.7 nm) at 35 K below melting. We use Classical Nucleation Theory (CNT) to estimate the ice-water interfacial free energy and the nucleation free-energy barrier. We obtain an interfacial free energy of 29(3) mN/m from an extrapolation of our results to the melting temperature. This value is in good agreement both with experimental measurements and with previous estimates from computer simulations of TIP4P-like models. Moreover, we obtain estimates of the nucleation rate from simulations of the critical cluster at the barrier top. The values we get for both models agree within statistical error with experimental measurements. At temperatures higher than 20 K below melting, we get nucleation rates slower than the appearance of a critical cluster in all water of the hydrosphere during the age of the universe. Therefore, our simulations predict that water freezing above this temperature must necessarily be heterogeneous.