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 moving group . V343 Normae comprises a K0 and mid-M star in a \sim 4.5 year orbit ( AaAb ) and a wide 10 \arcsec 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 \pm 0.10 M _ { \odot } and M _ { Ab } = 0.290 \pm 0.018 M _ { \odot } . 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 \pm 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 .