We measure the two-point correlation function xi(r) using a sample of 2219 galaxies in an area of 0.32 degrees^2 at z=0.7-1.35 from the first season of the DEEP2 Galaxy Redshift Survey. We find that xi(r) can be approximated by a power-law, xi(r)=(r/r_0)^-gamma, on scales 0.1-20 Mpc/h. In a sample with an effective redshift of z_eff=0.82, for a Lcdm cosmology we find r_0=3.53 +/-0.81 Mpc/h (comoving) and gamma=1.66 +/-0.12, while in a higher-redshift sample with z_eff=1.14 we find r_0=3.14 +/-0.72 Mpc/h and gamma=1.61 +/-0.11. We find that red, absorption-dominated, passively-evolving galaxies have a larger clustering scale length, r_0, and more prominent ``fingers of God'' than blue, emission-line, actively star-forming galaxies. Intrinsically brighter galaxies also cluster more strongly than fainter galaxies at z~1, with a significant luminosity-bias seen for galaxies fainter than M*. Our results are suggestive of evolution in the galaxy clustering within our survey volume and imply that the DEEP2 galaxies, with a median brightness one magnitude fainter than M* have an effective bias b=0.97 +/-0.13 if sigma_{8 DM}=1 today or b=1.20 +/-0.16 if sigma_{8 DM}=0.8 today. Given the strong luminosity-dependence in the bias that we measure at z~1, the galaxy bias at M* may be significantly greater. We note that our star-forming sample at z~1 has very similar selection criteria as 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 the clustering properties in the galaxy distribution seen in the local Universe were largely in place by z~1.