We measure the Milky Way's 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. We model the line-of-sight velocities of 3365 stars in 14 fields with b = 0° between 30° <= l <= 210° out to distances of 10 kpc using an axisymmetric kinematical model that includes a correction for the asymmetric drift of the warm tracer population (σ R ≈ 35 km s–1). We determine the local value of the circular velocity to be Vc (R 0) = 218 ± 6 km s–1 and find that the rotation curve is approximately flat with a local derivative between –3.0 km s–1 kpc–1 and 0.4 km s–1 kpc–1. We also measure the Sun's 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, ☉ = –10 ± 1 km s–1, and rotational velocity V phi, ☉ = 242+10 – 3 km s–1, in good agreement with local measurements of the Sun's 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–1 in Vc . Marginalizing over all the systematics that we consider, we find that Vc (R 0) < 235 km s–1 at >99 % confidence. We find an offset between the Sun's rotational velocity and the local circular velocity of 26 ± 3 km s–1, which is larger than the locally measured solar motion of 12 km s–1. This larger offset reconciles our value for Vc with recent claims that Vc >~ 240 km s–1. Combining our results with other data, we find that the Milky Way's dark-halo mass within the virial radius is ~8 × 1011 M ☉.