We calculate the dependence of the electrostatic persistence length, l_e, of weakly charged flexible polyelectrolyte chains using a self-consistent variational theory. The variation of l_e with kappa, the inverse Debye screening length, is controlled by the parameter l_0 l_B/A^2, where l_0 is the bare persistence length, l_B is the Bjerrum length, and A is the mean distance between charges along the chain. Several distinct regimes for the dependence of l_e on kappa emerge depending on the value of l_0 l_B/A^2. We show that when l_0 l_B /A^2 << 1 we recover the classical result, l_e propto kappa^{-2}. For intermediate values of l_0 l_B /A^2, l_e propto kappa^{-1}. In this regime one can also get l_e propto kappa^{-y} with y < 1 depending on the strength of the Coulomb interaction. Qualitative comparisons between our theory and simulations as well as other theories are presented.
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