HR 8799 hosts four directly imaged giant planets, but none has a mass measured from first principles. We present the first dynamical mass measurement in this planetary system, finding that the innermost planet HR~8799~e has a mass of $9.6^{+1.9}_{-1.8} , M_{rm Jup}$. This mass results from combining the well-characterized orbits of all four planets with a new astrometric acceleration detection (5$sigma$) from the Gaia EDR3 version of the Hipparcos-Gaia Catalog of Accelerations. We find with 95% confidence that HR~8799~e is below $13, M_{rm Jup}$, the deuterium-fusing mass limit. We derive a hot-start cooling age of $42^{+24}_{-16}$,Myr for HR~8799~e that agrees well with its hypothesized membership in the Columba association but is also consistent with an alternative suggested membership in the $beta$~Pictoris moving group. We exclude the presence of any additional $gtrsim$5-$M_{rm Jup}$ planets interior to HR~8799~e with semi-major axes between $approx$3-16,au. We provide proper motion anomalies and a matrix equation to solve for the mass of any of the planets of HR~8799 using only mass ratios between the planets.
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