Dissociation of diatomic molecules with odd number of electrons always causes the unpaired electron to localize on one of the two resulting atomic fragments. In the simplest diatomic molecule H2+ dissociation yields a hydrogen atom and a proton with the sole electron ending up on one of the two nuclei. That is equivalent to breaking of a chemical bond-the most fundamental chemical process. Here we observe such electron localization in real time by performing a pump-probe experiment. We demonstrate that in H2+ electron localization is complete in just 15 fs when the molecule's internuclear distance reaches 8 atomic units. The measurement is supported by a theoretical simulation based on numerical solution of the time-dependent Schrödinger equation. This observation advances our understanding of detailed dynamics of molecular dissociation.