  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. The new
wave function is symmetric under the exchange of particles and reproduces the
experimental equation of state. Results for the one-body density matrix show
the existence of off-diagonal long-range order with a very small condensate
fraction $\sim 10^{-4}$. The superfluid density of the commensurate system is
below our resolution threshold, $\rho_s/\rho < 10^{-5}$. With a 1%
concentration of vacancies the superfluid density is manifestly larger,
$\rho_s/\rho=3.2(1) \cdot 10^{-3}$.
