We present precise radial velocity observations of WASP-47, a star known to host a hot Jupiter, a distant Jovian companion, and, uniquely, two additional transiting planets in short-period orbits: a super-Earth in a ~19 hr orbit, and a Neptune in a ~9 day orbit. We analyze our observations from the HARPS-N spectrograph along with previously published data to measure the most precise planet masses yet for this system. When combined with new stellar parameters and reanalyzed transit photometry, our mass measurements place strong constraints on the compositions of the two small planets. We find that, unlike most other ultra-short-period planets, the inner planet, WASP-47 e, has a mass (6.83+/-0.66 M_{Earth}) and a radius (1.810+/-0.027 R{Earth}_) that are inconsistent with an Earth-like composition. Instead, WASP-47 e likely has a volatile-rich envelope surrounding an Earth-like core and mantle. We also perform a dynamical analysis to constrain the orbital inclination of WASP-47 c, the outer Jovian planet. This planet likely orbits close to the plane of the inner three planets, suggesting a quiet dynamical history for the system. Our dynamical constraints also imply that WASP-47 c is much more likely to transit than a geometric calculation would suggest. We calculate a transit probability for WASP-47 c of about 10%, more than an order of magnitude larger than the geometric transit probability of 0.6%.