Using the results of a high precision calculation of the solar neutrino survival probability for Earth crossing neutrinos in the case of the MSW nu(e) --> 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 the 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 B-8 and Be-7 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 studied. It is shown, in particular, that for sin(2) 2 theta V less than or equal to 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) greater than or similar to 10(-4) for only mantle crossing, core crossing, and (only mantle crossing+core crossing) neutrinos. The dependence of the D-N asymmetry in the three data samples on the recoil-e(-) energy threshold is also investigated. Our results indicate that the Super-Kamiokande experiment might be able to test the sin(2) 2 theta(V) less than or equal to 0.01 region of the MSW solution of the solar neutrino problem by performing selective D-N asymmetry measurements.