We measure the two-point correlation function ξ(rp, π) in a sample of 2219 galaxies between z = 0.7 and 1.35 to a magnitude limit of RAB = 24.1 from the first season of the DEEP2 Galaxy Redshift Survey. From ξ(rp, π) we recover the real-space correlation function, ξ(r), which we find can be approximated within the errors by a power law, ξ(r) = (r/r0)-γ, on scales ~0.1-10 h-1 Mpc. In a sample with an effective redshift of zeff = 0.82, for a ΛCDM cosmology we find r0 = 3.53 ± 0.81 h-1 Mpc (comoving) and γ = 1.66 ± 0.12, while in a higher redshift sample with zeff = 1.14 we find r0 = 3.12 ± 0.72 h-1 Mpc and γ = 1.66 ± 0.12. These errors are estimated from mock galaxy catalogs and are dominated by the cosmic variance present in the current data sample. We find that red, absorption-dominated, passively evolving galaxies have a larger clustering scale length, r0, than blue, emission-line, actively star-forming galaxies. Intrinsically brighter galaxies also cluster more strongly than fainter galaxies at z ≃ 1. Our results imply that the DEEP2 galaxies have an effective bias b = 0.96 ± 0.13 if σ8DM = 1 today or b = 1.19 ± 0.16 if σ8DM = 0.8 today. This bias is lower than that predicted by semianalytic simulations at z ≃ 1, which may be the result of our R-band target selection. We discuss possible evolutionary effects within our survey volume, and we compare our results with galaxy-clustering studies at other redshifts, noting that our star-forming sample at z ≃ 1 has selection criteria very similar to the Lyman break galaxies at z ≃ 3 and that our red, absorption-line sample displays a clustering strength comparable to the expected clustering of the Lyman break galaxy descendants at z ≃ 1. Our results demonstrate that galaxy-clustering properties as a function of color, spectral type, and luminosity seen in the local universe were largely in place by z ≃ 1.