  We have computed the one--body density matrix rho_1 in solid 4He at T=0 K
using the Shadow Wave Function (SWF) variational technique. The accuracy of the
SWF has been tested with an exact projector method. We find that off-diagonal
long range order is present in rho_1 for a perfect hcp and bcc solid 4He for a
range of densities above the melting one, at least up to 54 bars. This is the
first microscopic indication that Bose Einstein Condensation (BEC) is present
in perfect solid 4He. At melting the condensate fraction in the hcp solid is
5*10^{-6} and it decreases by increasing the density. The key process giving
rise to BEC is the formation of vacancy--interstitial pairs. We also present
values for Leggett's upper bound on the superfluid fraction deduced from the
exact local density.
