The elastic constants ${c}_{11}$, ${c}_{12}$, and ${c}_{44}$ of single-crystal strontium titanate have been measured as a function of temperature from 300 to 108\ifmmode^\circ\else\textdegree\fi{}K by determining the velocity of an ultrasonic wave. At 108\ifmmode^\circ\else\textdegree\fi{}K the strontium titanate undergoes a phase transition to a tetragonal structure which causes a marked change of the elastic properties and the appearance of a domain structure, but does not cause a discontinuity of the dielectric constant. The phase transition is free of hysteresis. The dependence of the elastic properties on a dc electric field parallel to the velocity of sound propagation was measured and found independent of the dielectric properties. The implications of this result are discussed.