  Using the results of a high precision calculation of the solar neutrino
survival probability for Earth crossing neutrinos in the case of MSW $\nu_e \to
\nu_{\mu(\tau)}$ transition solution of the solar neutrino problem, performed
in an earlier study, we derive predictions for the one-year averaged day-night
(D-N) asymmetry in the deformations of the $e^-$ - spectrum to be measured with
the Super - Kamiokande detector, and for the D-N asymmetry in the energy-
integrated one year signal in this detector. The asymmetries are calculated for
solar $\nu_e$ crossing the Earth mantle only, the core and the (mantle + core)
for a large representative set of values of the MSW transition parameters
$\Delta m^2$ and $\sin^2 2\theta_v$ from the ``conservative'' MSW solution
region obtained by taking into account possible uncertainties in the values of
the $^8$B and $^7$Be neutrino fluxes. The effect of the uncertainties in the
value of the bulk matter density and in the chemical composition of the core,
on the D-N asymmetry predictions is discussed. It is shown, in particular, that
for $\sin^2 2\theta_v \leq 0.013$ the one year average D-N asymmetry for
neutrinos crossing the Earth core can be larger than the asymmetry for (only
mantle crossing + core crossing) neutrinos by a factor of up to six. Iso -
(D-N) asymmetry contours in the $\Delta m^2 - \sin^2 2\theta_v$ plane for the
Super - Kamiokande detector are derived in the region $\sin^2 2\theta_v \gsim
10^{-4}$ for only mantle crossing, core crossing and (only mantle crossing +
core crossing) neutrinos. Our results indicate that the Super - Kamiokande
experiment might be able to test the $\sin^2 2\theta_v \le 0.01$ region of the
MSW solution of the solar neutrino problem by performing selective D-N
asymmetry measurements.
