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}Picb. We use the Markov Chain Monte Carlo orbit code orvara to fit 15years 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}Picb and 8.3{+/-}1.0M_Jup_ for {beta}Picc. 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}Picb, and an eccentricity of 0.21_-0.09_^+0.16^ for {beta}Picc, with the two orbital planes aligned to within ~0.5{deg}. Both planets' masses are within ~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 ~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 ~5 smaller than the uncertainties we derive here. Future GRAVITY observations will improve the constraints on {beta}Picc's mass and (especially) eccentricity, but improved constraints on the mass of {beta}Picb will likely require years of additional radial velocity monitoring and improved precision from future Gaia data releases.