In this paper, we describe a new, high-frequency, time-resolved scanning acoustic microscope developed for studying dynamical processes in biological cells. The new acoustic microscope operates in a time-resolved mode. The center frequency is 0.86 GHz, and the pulse duration is 5 ns. With such a short pulse, layers thicker than 3 microm can be resolved. For a cell thicker than 3 microm, the front echo and the echo from the substrate can be distinguished in the signal. Positions of the first and second pulses are used to determine the local impedance of the cell modeled as a thin liquid layer that has spatial variations in its elastic properties. The low signal-to-noise ratio in the acoustical images is increased for image generation by averaging the detected radio frequency signal over 10 measurements at each scanning point. In conducting quantitative measurements of the acoustic parameters of cells, the signal can be averaged over 2000 measurements. This approach enables us to measure acoustical properties of a single HeLa cell in vivo and to derive elastic parameters of subcellular structures. The value of the sound velocity inside the cell (1534.5 +/- 33.6 m/s) appears to be only slightly higher than that of the cell medium (1501 m/s).