We use silicon strip detectors (originally developed for the CLEO III high energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70,000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 <= R_lambda <= 970. Particle trajectories exhibiting accelerations up to 16,000 m\s^2 (40 times the rms value) are routinely observed. The probability density function of the acceleration is found to have Reynolds number dependent stretched exponential tails. The moments of the acceleration distribution are calculated. The scaling of the acceleration component variance with the energy dissipation is found to be consistent with the results for low Reynolds number direct numerical simulations, and with the K41 based Heisenberg-Yaglom prediction for R_lambda >= 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at R_lambda = 970. The coupling of the acceleration to the large scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time tau_eta.