We have performed the first experimental investigation of quantum interference corrections to the conductivity of a bilayer graphene structure. A negative magnetoresistance--a signature of weak localization--is observed at different carrier densities, including the electroneutrality region. It is very different, however, from the weak localization in conventional two-dimensional systems. We show that it is controlled not only by the dephasing time, but also by different elastic processes that break the effective time-reversal symmetry and provide intervalley scattering.