  In this manuscript, we continue with the systematic study of the speed of
extinction of continuous state branching processes in L\'evy environments under
more general branching mechanisms. Here, we deal with the weakly subcritical
regime under the assumption that the branching mechanism is regularly varying.
We extend recent results of Li and Xu [14] and Palau et al. [17], where it is
assumed that the branching mechanism is stable and complement the recent
articles of Bansaye et al. [2] and by the authors in [7], where the critical
and the strongly and intermediate subcritical cases were treated, respectively.
Our methodology combines a path analysis of the branching process together with
its L\'evy environment, fluctuation theory for L\'evy processes and the
asymptotic behaviour of exponential functionals of L\'evy processes. Our
approach is inspired by Afanasyev et al. [1], where the discrete analogue was
obtained, and by [2] and [7].
