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 + N0.837495⁺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 + N45.29485⁺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⁺¹'¹⁷ ₋₁.₂₉ M ⊕, R P = 1.416⁺⁰'⁰³³ ₋₀.₀₃₆ R ⊕, and ρP = 8.8⁺²'¹ ₋₂.₉ g cm⁻³. Kepler-10b is the smallest transiting exoplanet discovered to date.