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 MCMC orbit code orvara to fit fifteen 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 $9.3^{+2.6}_{-2.5}, M_{rm Jup}$ for $beta$ Pic b and $8.3pm 1.0,M_{rm Jup}$ for $beta$ Pic c. These masses are robust to modest changes to the input data selection. We find a well-constrained eccentricity of $0.119 pm 0.008$ for $beta$ Pic b, and an eccentricity of $0.21^{+0.16}_{-0.09}$ for $beta$ Pic c, with the two orbital planes aligned to within $sim$0.5$^circ$. Both planets masses are within $sim$1$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 $sim$5 times smaller than the GRAVITY uncertainties. Systematics in the masses and orbital parameters from orvaras approximate treatment of multiplanet orbits are a factor of $sim$5 smaller than the uncertainties we derive here. Future GRAVITY observations will improve the constraints on $beta$ Pic cs mass and (especially) eccentricity, but improved constraints on the mass of $beta$ Pic b will likely require years of additional RV monitoring and improved precision from future Gaia data releases.
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