The single-crystal elastic constants of SiO2, coesite (monoclinic, C2/c), have been determined from Brillouin scattering measurements on a 100-μm crystal. The elastic constants inferred from acoustic velocities for 80 different propagation directions are (in units of Mbar) c11=1.608, c22=2.304, c33=2.316, c44=0.678, c55=0.733, c66=0.588, c12=0.821, c13=1.029, c15=-0.362, c23=0.356, c25=0.026, c35=-0.393, and c46=0.099. This experimental technique provides a method of determining single-crystal elastic properties for materials such as high-pressure phases that are not available in sufficiently large sizes for ultrasonic measurements.