We consider the seesaw mechanism of neutrino mass generation in the light o
f our present knowledge of the neutrino masses and mixing. We analyse the s
eesaw mechanism constrained by the following assumptions: (1) minimal seesa
w with no Higgs triplets, (2) hierarchical Dirac masses of neutrinos, (3) l
arge lepton mixing primarily or solely due to the mixing in the right-hande
d neutrino sector, and (4) unrelated Dirac and Majorana sectors of neutrino
masses. We show that large mixing governing the dominant channel of the at
mospheric neutrino oscillations can be naturally obtained and point out tha
t this constrained seesaw mechanism favours the normal mass hierarchy for t
he light neutrinos leading to a small V-e3 entry of the lepton mixing matri
x and a mass scale of the lightest right handed neutrino M similar or equal
to 10(10)-10(11) GeV. Any of the three main neutrino oscillation solutions
to the solar neutrino problem can be accommodated. The inverted mass hiera
rchy and quasi-degeneracy of neutrinos are disfavoured in our scheme. (C) 2
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