  We consider the seesaw mechanism of neutrino mass generation in the light of
our present knowledge of the neutrino masses and mixing. We analyse the seesaw
mechanism constrained by the following assumptions: (1) minimal seesaw with no
Higgs triplets, (2) hierarchical Dirac masses of neutrinos, (3) large lepton
mixing primarily or solely due to the mixing in the right-handed neutrino
sector, and (4) unrelated Dirac and Majorana sectors of neutrino masses. We
show that large mixing governing the dominant channel of the atmospheric
neutrino oscillations can be naturally obtained and point out that this
constrained seesaw mechanism favours the normal mass hierarchy for the light
neutrinos leading to a small $V_{e3}$ entry of the lepton mixing matrix and a
mass scale of the lightest right handed neutrino $M\simeq 10^{10} - 10^{11}$
GeV. Any of the three main neutrino oscillation solutions to the solar neutrino
problem can be accommodated. The inverted mass hierarchy and quasi-degeneracy
of neutrinos are disfavoured in our scheme.
