  We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor
in the large $x$ asymptotics of the Painlev\'e III tau-function. Our result
proves the conjectural formula for this pre-factor obtained recently by O.
Lisovyy, Y. Tykhyy, and the first co-author with the help of the recently
discovered connection of the Painlev\'e tau-functions with the Virasoro
conformal blocks. Our approach does not use this connection, and it is based on
the Riemann-Hilbert method.
