A theory of weak localization in graphene with Rashba splitting of the energy spectrum is developed. Anomalous magnetoresistance caused by weak localization is calculated accounting for inter- and intravalley, and spin-orbit and spin-valley scattering processes. It is shown that the anomalous magnetoresistance is described by an expression different from the traditional Hikami-Larkin-Nagaoka formula. The reason is that the effect of Rashba splitting gives rise to the spin-orbit vector potential which is not reduced to a spin dephasing only. The developed theory can be applied to heterostructures of graphene with transition metal dichalcogenides.