We present the results of the asteroseismic analysis of the red-giant star KIC 4351319 (TYC 3124-914-1), observed for 30 days in short-cadence mode with the Kepler satellite. The analysis has allowed us to determine the large and small frequency separations, and the frequency of maximum oscillation power. The high signal-to-noise ratio of the observations allowed us to identify 25 independent pulsation modes whose frequencies range approximately from 300 to 500 muHz. The observed oscillation frequencies together with the accurate determination of the atmospheric parameters (effective temperature, gravity and metallicity), provided by additional ground-based spectroscopic observations, enabled us to theoretically interpret the observed oscillation spectrum. KIC 4351319 appears to oscillate with a well defined solar-type p-modes pattern due to radial acoustic modes and non-radial nearly pure p modes. In addition, several non-radial mixed modes have been identified. Theoretical models well reproduce the observed oscillation frequencies and indicate that this star, located at the base of the ascending red-giant branch, is in the hydrogen-shell burning phase, with a mass of about 1.3 solar masses, a radius of about 3.4 solar radii and an age of about 5.6 Gyr. The main parameters of this star have been determined with an unprecedent level of precision for a red-giant star, with uncertainties of 2% for mass, 7% for age, 1% for radius, and 4% for luminosity.