  We show that in a superconductor where two or more bands cross the Fermi
level it is possible, in the framework of the conventional (s-wave) BCS theory,
that the sign of the superconducting gap is different on the different sheets
of the Fermi surface. At least one of two conditions has to be satisfied: (1)
The interband pairing interaction is weaker than the Coulomb pseudopotential,
while the intraband one is stronger, or (2) there is strong interband
scattering by magnetic impurities. In the case of YBa$_2$% Cu$_3$O$_7$ we shall
argue that the first condition is possibly satisfied, and the second one very
likely satisfied. In many aspects such a sign-reversal $s-$wave superconductor
is similar to a d-wave superconductor, and thus demands revising recent
experiments aimed to distinguish between the $s-$ and $d-$wave
superconductivity in this compound.
