We calibrate the galaxy cluster mass - temperature relation using the temperature profiles of intracluster gas observed with ASCA (for hot clusters) and ROSAT (for cool groups). Our sample consists of apparently relaxed clusters for which the total masses are derived assuming hydrostatic equilibrium. The sample provides data on cluster X-ray emission-weighted cooling flow-corrected temperatures and total masses up to r_1000. The resulting M-T scaling in the 1-10 keV temperature range is M_1000 = (1.23 +- 0.20)/h_50 10^15 Msun (T/10 keV)^{1.79 +- 0.14} with 90% confidence errors, or significantly (99.99% confidence) steeper than the self-similar relation M propto T^{3/2}. For any given temperature, our measured mass values are significantly smaller compared to the simulation results of Evrard et al. (1996) that are frequently used for mass-temperature scaling. The higher-temperature subsample (kT > 4 keV) is consistent with M propto T^{3/2}, allowing the possibility that the self-similar scaling breaks down at low temperatures, perhaps due to heating by supernovae that is more important for low-temperature groups and galaxies as suggested by earlier works.