  One of the outstanding issues in neutrino physics is the experimental
determination of the neutrino mass hierarchy: Is the order of the neutrino
masses ``normal'' - m_1^2<m_2^2<m_3^2 - or is it inverted - m_3^2<m_1^2<m_2^2,
with m_2^2-m_1^2 << m_2^2,m_1^2? While this issue can be resolved in
next-generation long-baseline nu_{mu} to nu_e neutrino oscillation studies if
|U_{e3}|^2 is large enough, a clear strategy on how to resolve it if |U_{e3}|^2
is sufficiently small is still lacking. We study the capability of
non-oscillation probes of neutrino masses to determine the neutrino mass
ordering. We concentrate on studies of m_{nu_e}, the kinematical neutrino mass
to which precise studies of tritium beta-decay are sensitive, m_{ee}, the
effective mass to which the rate for neutrinoless double-beta decay is
sensitive if the neutrinos are Majorana fermions, and Sigma, the sum of the
neutrino masses, to which cosmological probes of the energy budget of the
Universe are sensitive. We find that combined measurements of m_{ee}, Sigma,
and m_{nu_e} are capable of establishing the neutrino mass hierarchy if these
measurements are precise enough and if one ``gets lucky.'' We quantify the
previous sentence in detail by performing a numerical analysis of a large
number of theoretical data sets, for different measured values of m_{nu_e},
m_{ee}, and Sigma, keeping in mind the ultimate sensitivity that can be reached
by next (and next-to-next) generation experiments.
