NASA's Kepler Mission uses transit photometry to determine the frequency of Earth-size planets in or near the habitable zone of Sun-like stars. The mission reached a milestone toward meeting that goal: the discovery of its first rocky planet, Kepler-10b. Two distinct sets of transit events were detected: (1) a 152 ± 4 ppm dimming lasting 1.811 ± 0.024 hr with ephemeris T [BJD] = 2454964.57375^(+0.00060)_(–0.00082) + N * 0.837495^(+0.000004)_(–0.000005) days and (2) a 376 ± 9 ppm dimming lasting 6.86 ± 0.07 hr with ephemeris T [BJD] = 2454971.6761^(+0.0020)_(–0.0023) + N * 45.29485^(+0.00065) _(–0.00076) days. Statistical tests on the photometric and pixel flux time series established the viability of the planet candidates triggering ground-based follow-up observations. Forty precision Doppler measurements were used to confirm that the short-period transit event is due to a planetary companion. The parent star is bright enough for asteroseismic analysis. Photometry was collected at 1 minute cadence for >4 months from which we detected 19 distinct pulsation frequencies. Modeling the frequencies resulted in precise knowledge of the fundamental stellar properties. Kepler-10 is a relatively old (11.9 ± 4.5 Gyr) but otherwise Sun-like main-sequence star with T_(eff) = 5627 ± 44 K, M_⋆ = 0.895 ± 0.060 M_⊙ , and R_⋆ = 1.056 ± 0.021 R_⊙. Physical models simultaneously fit to the transit light curves and the precision Doppler measurements yielded tight constraints on the properties of Kepler-10b that speak to its rocky composition: M_P = 4.56^9+1.17)_(–1.29) M_⊕, R_P = 1.416^(+0.033)_(–0.036) R_⊕, and ρ_P = 8.8^(+2.1)_(–2.9) g cm^(–3). Kepler-10b is the smallest transiting exoplanet discovered to date