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) much less than 1 we recover the classical result, l(e)(proportional to)kappa(-2). For intermediate values of l(0)l(B)/A(2), l(e)(proportional to)kappa(-1). In this regime one can also get l(e)(proportional to)kappa(-y) with y<1 depending on the strength of the Coulomb interaction. When l(0)l(B)/A(2)much greater than 1 we find l(e)(proportional to)kappa(-2). Qualitative comparisons between our theory and simulations as well as other theories are presented.