  We present general conditions on Dirac and Majorana mass terms under which a
type-I seesaw mechanism can lead to three exactly massless neutrinos at the
tree level. We depict several examples where the conditions are satisfied and
relate some of them to an underlying U(1) symmetry. We show that higher order
corrections may generate the small observed masses and this may be achieved
even when the heavy Majorana neutrinos are at the electroweak scale or a little
higher.
