  Generic c=1 four-point conformal blocks on the Riemann sphere can be seen as
the coefficients of Fourier expansion of the tau function of Painlev\'e VI
equation with respect to one of its integration constants. Based on this
relation, we show that c=1 fusion matrix essentially coincides with the
connection coefficient relating tau function asymptotics at different critical
points. Explicit formulas for both quantities are obtained by solving certain
functional relations which follow from the tau function expansions. The final
result does not involve integration and is given by a ratio of two products of
Barnes G-functions with arguments expressed in terms of conformal
dimensions/monodromy data. It turns out to be closely related to the volume of
hyperbolic tetrahedron.
