  If future neutrino oscillation experiments show that the neutrino mass
spectrum is with normal ordering, m1 < m2 < m3, and the searches for
neutrinoless double beta-decay with sensitivity to values of the effective
Majorana mass |<m>| > 0.01 eV give negative results, the next frontier in the
quest for neutrinoless double beta-decay will correspond to |<m>| ~ 0.001 eV.
Assuming that massive neutrinos are Majorana particles and their exchange is
the dominant mechanism generating neutrinoless double beta-decay, we analise
the conditions under which |<m>|, in the case of three neutrino mixing and
neutrino mass spectrum with normal ordering, would satisfy |<m>| > 0.001 eV. We
consider the specific cases of i) normal hierarchical neutrino mass spectrum,
ii) of relatively small value of the CHOOZ angle theta13 as well as iii) the
general case of spectrum with normal ordering, partial hierarchy and a value of
theta13 close to the existing upper limit. We study the ranges of the lightest
neutrino mass m1 and/or of sin^2 theta13, for which |<m>|> 0.001 eV and discuss
the phenomenological implications of such scenarios. We provide also an
estimate of |<m>| when the three neutrino masses and the neutrino mixing
originate from neutrino mass term of Majorana type for the (left-handed)
flavour neutrinos and m1 Ue1^2 + m2 U_e2^2 + m3 Ue3^2 =0, but there does not
exist a symmetry which forbids the neutrinoless double beta-decay.
