  We derive exact closed form expressions for the first few terms of the
short-distance Taylor expansion of the one-body correlation function of the
Lieb-Liniger gas. As an intermediate result we obtain the high-p asymptotics of
the momentum distribution of both free and harmonically trapped atoms and show
that it obeys a universal 1/p^4 law for_all_ values of the interaction
strength. We discuss the ways to observe the predicted momentum distributions
experimentally, regarding them as a sensitive identifier for the
Tonks-Girardeau regime of strong correlations.
