  The K-electron capture half-life of $^{123}$Te has been recently measured to
be $t^K_{exp}=2.4\times 10^{19}$ yr, and constitutes the longest half-life ever
measured in a single $\beta$-transition of any nuclear species. We have
calculated this second unique forbidden transition within the framework of the
proton-neutron quasi-particle random phase approximation, making use of
Skyrme-type effective interactions. A strong cancellation effect between
particle-hole and particle-particle correlations is found. The model, without
any renormalization of the force, provides a lower limit for the K-electron
capture half-life of $\sim 10^{17}$ yr, which unambiguously rules out the old
experimental values of $10^{13} - 10^{14}$ yr. A few percent increase of the
particle-particle matrix elements of the Skyrme interaction allows to reproduce
the experimental findings.
