  We present a new method for deriving cosmological constraints based on the
evolution of the baryon mass function of galaxy clusters, and implement it
using 17 distant clusters from our 160deg2 ROSAT survey. The method uses the
cluster baryon mass as a proxy for the total mass, thereby avoiding the large
uncertainties of the M_tot-T or M_tot-L_X relations used in all previous
studies. Instead, we rely on a well-founded assumption that the M_b/M_tot ratio
is a universal quantity, which should result in a much smaller systematic
uncertainty. Taking advantage of direct and accurate Chandra measurements of
the gas masses for distant clusters, we find strong evolution of the baryon
mass function between z>0.4 and the present. The observed evolution defines a
narrow band in the Omega_m-Lambda plane, Omega_m + 0.23Lambda = 0.41+-0.10 at
68% confidence, which intersects with constraints from the Cosmic Microwave
Background and supernovae Ia near Omega_m=0.3 and Lambda=0.7.
