We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particle at a turbulent Reynolds number R(lambda) = 740, with two decades of time resolution, below the Lagrangian correlation time. We observe that the Lagrangian velocity spectrum has a Lorentzian form E(L)(omega) = u(2)(rms)T(L)/[1+(T(L)omega)(2)], in agreement with a Kolmogorov-like scaling in the inertial range. The probability density functions of the velocity time increments display an intermittency which is more pronounced than that of the corresponding Eulerian spatial increments.