We measure the Milky Ways rotation curve over the Galactocentric range 4 kpc <~ R <~ 14 kpc from the first year of data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE). We model the line-of-sight velocities of 3,365 stars in fourteen fields with b = 0 deg between 30 deg < l < 210 deg out to distances of 10 kpc using an axisymmetric kinematical model that includes a correction for the asymmetric drift of the warm tracer population (sigma_R ~ 35 km/s). We determine the local value of the circular velocity to be V_c(R_0) = 218 +/- 6 km/s and find that the rotation curve is approximately flat with a local derivative between -3.0 km/s/kpc and 0.4 km/s/kpc. We also measure the Suns position and velocity in the Galactocentric rest frame, finding the distance to the Galactic center to be 8 kpc < R_0 < 9 kpc, radial velocity V_{R,sun} = -10 +/- 1 km/s, and rotational velocity V_{phi,sun} = 242^{+10}_{-3} km/s, in good agreement with local measurements of the Suns radial velocity and with the observed proper motion of Sgr A*. We investigate various systematic uncertainties and find that these are limited to offsets at the percent level, ~2 km/s in V_c. Marginalizing over all the systematics that we consider, we find that V_c(R_0) < 235 km/s at >99% confidence. We find an offset between the Suns rotational velocity and the local circular velocity of 26 +/- 3 km/s, which is larger than the locally-measured solar motion of 12 km/s. This larger offset reconciles our value for V_c with recent claims that V_c >~ 240 km/s. Combining our results with other data, we find that the Milky Ways dark-halo mass within the virial radius is ~8x10^{11} M_sun.
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