We present gas mass fractions of 38 massive galaxy clusters at redshifts 0.14 ≤ z ≤ 0.89, derived from Chandra X-ray and OVRO/BIMA interferometric Sunyaev-Zel'dovich effect (SZE) measurements. We use three models for the gas distribution: (1) an isothermal β-model fit jointly to the X-ray data at radii beyond 100 kpc and to all of the SZE data, (2) a nonisothermal double β-model in hydrostatic equilibrium with a Navarro-Frenk-White (NFW) dark matter distribution, fit jointly to all of the X-ray and SZE data, and (3) an isothermal β-model fit only to the SZE spatial data. We show that the isothermal model well characterizes the intracluster medium outside the cluster core and provides good fits to clusters with a range of morphological properties. X-ray and SZE mean gas mass fractions for model 1 are f_gas(X-ray) = 0.110 ± 0.003 and f_gas(SZE) = 0.116 ± 0.005 assuming = ; uncertainties are statistical followed by systematic at 68% confidence. For model 2, f_gas(X-ray) = 0.119 ± 0.003 and f_gas(SZE) = 0.121 ± 0.005. For model 3, f_gas(SZE) = 0.120 ± 0.009. The agreement in the results shows that the core can be accounted for satisfactorily by either excluding it from fits to the X-ray data or modeling the intracluster gas with a nonisothermal double β-model. We find that the SZE is largely insensitive to core structure. Our results indicate that the ratio of gas mass fraction within r₂₅₀₀ to the cosmic baryon fraction, f_gas/ , is 0.68, where statistical and systematic uncertainties are included at 68% confidence. By assuming that cluster gas mass fractions are independent of redshift, we find that the results agree with standard ΛCDM cosmology and are inconsistent with a flat matter-dominated (Ω_M = 1) universe.