  We use the relation between the volumes of the strata of meromorphic
quadratic differentials with at most simple poles on the Riemann sphere and
counting functions of the number of (bands of) closed geodesics in associated
flat metrics with singularities to prove a very explicit formula for the volume
of each such stratum conjectured by M. Kontsevich a decade ago.
  The proof is based on a formula for the Lyapunov exponents of the geodesic
flow on the moduli space, which gives a recursive formula from which the
volumes could be recovered.
  An appendix with an ergodic theorem proved by Jon Chaika is added to the
current version. Applying this ergodic theorem to the Teichmueller geodesic
flow we obtain EXACT quadratic asymptotics for the number of (bands of) closed
trajectories and for the number of generalized diagonals in almost all
right-angled billiards. All coefficients in the asymptotics (expressed in terms
of the associated Siegel-Veech constants) are explicitly computed.
