  This is the first of two articles aimed at providing comprehensive
predictions for the day-night (D-N) effect for the Super-Kamiokande detector in
the case of the MSW $\nu_e \to \numt$ transition solution of the solar neutrino
problem. The one-year averaged probability of survival of the solar $\nue$
crossing the Earth mantle, the core, the inner 2/3 of the core, and the (core +
mantle) is calculated with high precision (better than 1%) using the elliptical
orbit approximation (EOA) to describe the Earth motion around the Sun. Results
for the survival probability in the indicated cases are obtained for a large
set of values of the MSW transition parameters $\Delta m^2$ and
$sin^22\theta_{V}$ from the ``conservative'' regions of the MSW solution,
derived by taking into account possible relatively large uncertainties in the
values of the $^{8}$B and $^{7}$Be neutrino fluxes. Our results show that the
one-year averaged D-N asymmetry in the $\nu_e$ survival probability for
neutrinos crossing the Earth core can be, in the case of $sin^22 \theta_{V}
\leq 0.13$, larger than the asymmetry in the probability for (only mantle
crossing + core crossing) neutrinos by a factor of up to six. The enhancement
is larger in the case of neutrinos crossing the inner 2/3 of the core. This
indicates that the Super-Kamiokande experiment might be able to test the
$sin^22\theta_{V} \leq 0.01$ region of the MSW solution of the solar neutrino
problem by performing selective D-N asymmetry measurements.
