  It is shown that the $\nu_2 \to \nu_{e}$ and $\nu_{\mu} \to \nu_{e}$ ($\nu_e
\to \nu_{\mu (\tau)}$) transitions respectively of the solar and atmospheric
neutrinos in the Earth in the case of $\nu_e - \nu_{\mu (\tau)}$ mixing in
vacuum, are strongly enhanced by a new type of resonance when the neutrinos
cross the Earth core. The resonance is operative at small mixing angles but
differs from the MSW one. It is in many respects similar to the electron
paramagnetic resonance taking place in a specific configuration of two magnetic
fields. The conditions for existence of the new resonance include, in
particular, specific constraints on the neutrino oscillation lengths in the
Earth mantle and in the Earth core, thus the resonance is a ``neutrino
oscillation length resonance''. It leads also to enhancement of the $\nu_2 \to
\nu_e$ and $\nu_e \to \nu_s$ transitions in the case of $\nu_e - \nu_s$ mixing
and of the $\bar{\nu}_{\mu} \to \bar{\nu}_{s}$ (or $\nu_{\mu} \to \nu_{s}$)
transitions at small mixing angles. The presence of the neutrino oscillation
length resonance in the transitions of solar and atmospheric neutrinos
traversing the Earth core has important implications for current and future
solar and atmospheric neutrino experiments, and more specifically, for the
interpretation of the results of the Super-Kamiokande experiment.
