We present new astrometry for the young (12-21 Myr) exoplanet beta Pictoris b taken with the Gemini/NICI and Magellan/MagAO instruments between 2009 and 2012. The high dynamic range of our observations allows us to measure the relative position of beta Pic b with respect to its primary star with greater accuracy than previous observations. Based on a Markov Chain Monte Carlo analysis, we find the planet has an orbital semi-major axis of 9.1(-0.5)(+5.3) AU and orbital eccentricity <0.15 at 68% confidence (with 95% confidence intervals of 8.2-48 AU and 0.00-0.82 for semi-major axis and eccentricity, respectively, due to a long narrow degenerate tail between the two). We find that the planet has reached its maximum projected elongation, enabling higher precision determination of the orbital parameters than previously possible, and that the planet's projected separation is currently decreasing. With unsaturated data of the entire beta Pic system (primary star, planet, and disk) obtained thanks to NICI's semi-transparent focal plane mask, we are able to tightly constrain the relative orientation of the circumstellar components. We find the orbital plane of the planet lies between the inner and outer disks: the position angle (P.A.) of nodes for the planet's orbit (211.8 +/- 0 degrees.3) is 7.4 sigma greater than the P. A. of the spine of the outer disk and 3.2 sigma less than the warped inner disk P. A., indicating the disk is not collisionally relaxed. Finally, for the first time we are able to dynamically constrain the mass of the primary star beta Pic to 1.76(-0.17)(+0.18) M-circle dot.