The ground state of solid 4 He is studied using the diffusion Monte Carlo method and a new trial wave function able to describe the supersolid. This wave function is symmetric under the exchange of particles and used as a guiding function in the method allows for reproducing the experimental equation of state. The use of this zero-temperature technique overcomes the conceptual ambiguity of finite-temperature methods in the search of a supersolid. Results for the one-body density matrix show the existence of off-diagonal long-range order with a very small condensate fraction∼ 10-4, the specific value being not fully independent of the trial wave function due to the remaining bias in the extrapolated estimator. The superfluid density of the commensurate system is below our resolution threshold, ρ s/ρ< 10-5. This zero-temperature result is incompatible with recent experimental measures of superfluidity in solid 4 He showing that the origin of the experimental findings is not that of a supersolid in a perfect crystal. Introducing in the system a 1% concentration of vacancies the superfluid density is manifestly larger, ρ s/ρ= 3.2 (1)× 10-3.