We use compiled high-precision pulsar timing measurements to directly measure the Galactic acceleration of binary pulsars relative to the Solar System barycenter. Given the vertical accelerations, we use the Poisson equation to derive the Oort limit, i.e., the total volume mass density in the Galactic mid-plane. Our best-fitting model gives an Oort limit of $0.08^{0.05}_{-0.02} M_{odot}/rm pc^{3}$, which is close to estimates from recent Jeans analyses. Given the accounting of the baryon budget from McKee et al. (2015), we obtain a local dark matter density of $-0.004^{0.05}_{-0.02}~M_{odot}/rm pc^{3}$, which is slightly below other modern estimates but consistent within the current uncertainties of our method. While this first measurement of the Oort limit (and other Galactic parameters) has error bars that are currently several times larger than kinematical estimates, they should improve in the future. We also constrain the oblateness of the potential, finding it consistent with that expected from the disk and inconsistent with a potential dominated by a spherical halo, as is appropriate for our sample which is within a $sim$ kpc of the Sun. We find that the slope of the rotation curve is not constrained by current measurements of binary pulsar accelerations. We give a fitting function for the vertical acceleration $a_{z}$: $a_{z} = -alpha_{1}z$; $log_{10} (alpha_{1}/{rm Gyr}^{-2})=3.69^{0.19}_{-0.12}$. By analyzing interacting simulations of the Milky Way, we find that large asymmetric variations in $da_{z}/dz$ as a function of vertical height may be a signature of sub-structure. We end by discussing the power of combining constraints from pulsar timing and high-precision radial velocity (RV) measurements towards lines-of-sight near pulsars, to test theories of gravity and constrain dark matter sub-structure.
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