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 beta Pictoris (beta Pic) moving group. V343 Normae comprises a K0 and mid-M star in a similar to 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 M-Aa = 1.10 +/- 0.10M(circle dot) and M-Ab= 0.290 +/- 0.018 M-circle dot. 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 beta Pic moving group. We derive a model-dependent age for the beta 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.