  In the context of three-neutrino oscillations, we study the possibility of
using antineutrinos from nuclear reactors to explore the 10^{-4} {\rm eV^2} <
\ms \ltap 8\times 10^{-4} {\rm eV^2} region of the LMA MSW solution of the
solar neutrino problem and measure $\ms$ with high precision. The KamLAND
experiment is not expected to determine $\ms$ if the latter happens to lie in
the indicated region. By analysing both the total event rate suppression and
the energy spectrum distortion caused by \bar{\nu}_e oscillations in vacuum, we
show that the optimal baseline of such an experiment is L \sim (20 - 25) km.
Furthermore, for 10^{-4} {\rm eV^2} < \ms \ltap 5\times 10^{-4} {\rm eV^2}, the
same experiment might be used to try to distinguish between the two possible
types of neutrino mass spectrum - with normal or with inverted hierarchy, by
exploring the effect of interference between the atmospheric- and solar- \Delta
m^2 driven oscillations; for larger values of \ms not exceeding 8.0\times
10^{-4} eV^2, a shorter baseline, L \cong 10 km, would be needed for the
purpose. The indicated interference effect modifies in a characteristic way the
energy spectrum of detected events. Distinguishing between the two types of
neutrino mass spectrum requires, however, a high precision determination of the
atmospheric \Delta m^2, a sufficiently large \sin^2\theta and a non-maximal
\sin^22\theta_{\odot}, where \theta and \theta_{\odot} are the mixing angles
respectively limited by the CHOOZ and Palo Verde data and characterizing the
solar neutrino oscillations. It also requires a relatively high precision
measurement of the positron spectrum in the reaction \bar{\nu}_e + p \to e^{+}
+ n.
