-decay experiments could provide this information through the value of found [@PPSNO2bb].

The value of $m_1$ can be measured by the KATRIN experiment [@KATRIN] if the neutrino mass spectrum is of the QD type. In this case $m_1 \cong m_2  \cong m_3 \cong m_{\bar{\nu}_e}$. Given the currently allowed regions of values of and (see further), we have QD spectrum for $m_{1,2,3} \cong m_{\bar{\nu}_e} > 0.20$ eV. The KATRIN detector is designed to have a 1 s.d. error of 0.08 eV$^2$ on a measured value of $m_{\bar{\nu}_e}^2$. The most stringent upper limit on $m_{\bar{\nu}_e} \cong m_1$, which can be reached in this experiment, is 0.35 eV. The KATRIN experiment is expected to start in 2007.

The sum of neutrino masses $\Sigma$ can be determined by using data on the weak lensing of galaxies by large scale structure, and data on the cosmic microwave background (CMB) from the MAP and PLANCK experiments, with an estimated (1 s.d.) error of 0.04 eV [@Hu99]. The latter represents the estimated best precision that can possibly be achieved in the cosmological determination of $\Sigma$. If only an upper limit of $\sim (0.10 - 0.15)$ eV on $\Sigma$ will be obtained, that would strongly disfavor (if not 