  Using 482 pb$^{-1}$ of data taken at $\sqrt{s}=4.009$ GeV, we measure the
branching fractions of the decays of $D^{*0}$ into $D^0\pi^0$ and $D^0\gamma$
to be $\BR(D^{*0} \to D^0\pi^0)=(65.5\pm 0.8\pm 0.5)%$ and $\BR(D^{*0} \to
D^0\gamma)=(34.5\pm 0.8\pm 0.5)%$ respectively, by assuming that the $D^{*0}$
decays only into these two modes. The ratio of the two branching fractions is
$\BR(D^{*0} \to D^0\pi^0)/\BR(D^{*0} \to D^0\gamma) =1.90\pm 0.07\pm 0.05$,
which is independent of the assumption made above. The first uncertainties are
statistical and the second ones systematic. The precision is improved by a
factor of three compared to the present world average values.
