We present a comprehensive orbital analysis to the exoplanets beta Pictoris b and c that resolves previously reported tensions between the dynamical and evolutionary mass constraints on beta Pic b. We use the Markov Chain Monte Carlo orbit code orvara to fit 15 years of radial velocities and relative astrometry (including recent GRAVITY measurements), absolute astrometry from Hipparcos and Gaia, and a single relative radial velocity measurement between beta Pic A and b. We measure model-independent masses of 9.3(-2.5)(+2.6) M-Jup for beta Pic b and 8.3 1.0 M-Jup for beta Pic c. These masses are robust to modest changes to the input data selection. We find a well-constrained eccentricity of 0.119 0.008 for beta Pic b, and an eccentricity of 0.21(-0.09)(+0.16) beta Pic c, with the two orbital planes aligned to within similar to 05. Both planets' masses are within similar to 1 sigma of the predictions of hot-start evolutionary models and exclude cold starts. We validate our approach on N-body synthetic data integrated using REBOUND. We show that orvara can account for three-body effects in the beta Pic system down to a level similar to 5 times smaller than the GRAVITY uncertainties. Systematics in the masses and orbital parameters from orvara's approximate treatment of multiplanet orbits are a factor of similar to 5 smaller than the uncertainties we derive here. Future GRAVITY observations will improve the constraints on beta Pic c's mass and (especially) eccentricity, but improved constraints on the mass of beta Pic b will likely require years of additional radial velocity monitoring and improved precision from future Gaia data releases.