We present new spatially resolved astrometry and photometry from the Gemini Planet Imager of the inner binary of the young multiple star system V343 Normae, which is a member of the β Pictoris (β Pic) moving group. V343 Normae comprises a K0 and mid-M star in a ∼4.5 year orbit (AaAb) and a wide 10″ M5 companion (B). By combining these data with archival astrometry and radial velocities we fit the orbit and measure individual masses for both components of MAa = 1.10 ± 0.10 M⊙ and MAb = 0.290 ± 0.018 M⊙. Comparing to theoretical isochrones, we find good agreement for the measured masses and JHK band magnitudes of the two components consistent with the age of the β Pic moving group. We derive a model-dependent age for the β Pic moving group of 26 ±3 Myr by combining our results for V343 Normae with literature measurements for GJ 3305, which is another group member with resolved binary components and dynamical masses
