  We investigate the cross-over from three to one dimension in a Bose gas
confined in highly anisotropic traps. By using Quantum Monte-Carlo techniques,
we solve the many-body Schrodinger equation for the ground state and obtain
exact results for the energy per particle and the mean square radii of the
cloud in the transverse and longitudinal direction. Results are compared with
the predictions of mean-field theory obtained from the Gross-Pitaevskii
equation and with the 1D Lieb-Liniger equation of state. We explicitly prove
the occurrence of important beyond mean-field effects, including the appearance
of Fermi-like properties as the system enters the Tonks gas regime.
