The 13 adiabatic elastic stiffness moduli of natural muscovite have been measured using Brillouin scattering. Following the standard Institute of Radio Engineers convention, these contants are (in gigapascals) C11=181.0¿1.2, C22=178.4¿1.3, C33=58.6¿0.6, C44=16.5¿0.6, C55=19.5¿0.5, C66=72.0¿0.7, C23=21.2¿1.8, C13=25.6¿1.5, C12=48.8¿2.5, C15=-14.2¿0.8, C25=1.1¿3.7, C35=1.0¿0.6, and C46=-5.2¿0.9. The main observed features are the very great acoustic anisotropy due to the weak interlayer bonding and the small, but not insignificant, deviation from hexagonal symmetry (monoclinicity) in the velocity patterns.