Recent radiative lifetime measurements accurate to +/- 5% using laser-induced fluorescence (LIF) on 43 even-parity and 15 odd-parity levels of Ce II have been combined with new branching fractions measured using a Fourier transform spectrometer (FTS) to determine transition probabilities for 921 lines of Ce II. This improved laboratory data set has been used to determine a new solar photospheric Ce abundance, log epsilon = 1.61 +/- 0.01 (sigma = 0.06 from 45 lines), a value in excellent agreement with the recommended meteoritic abundance, log epsilon = 1.61 +/- 0.02. Revised Ce abundances have also been derived for the r-process-rich metal-poor giant stars BD+17 degrees 3248, CS 22892-052, CS 31082-001, HD 115444, and HD 221170. Between 26 and 40 lines were used for determining the Ce abundance in these five stars, yielding a small statistical uncertainty of +/- 0.01 dex similar to the solar result. The relative abundances in the metal-poor stars of Ce and Eu, a nearly pure r-process element in the Sun, matches r-process-only model predictions for solar system material. This consistent match with small scatter over a wide range of stellar metallicities lends support to these predictions of elemental fractions. A companion paper includes an interpretation of these new precision abundance results for Ce as well as new abundance results and interpretation for Pr, Dy, and Tm.