We present simulations of the basic model of weakly charged polyelectrolyte chains of variable intrinsic stiffness within the Debye - Hückel approximation. For intrinsically flexible chains the persistence length shows a sublinear dependence on the screening length in strong contrast to all known analytical approaches which propose an effective exponent of either y = 2 or y = 1. The observed exponent y varies as a function of the system parameters. With increasing intrinsic stiffness the corresponding effective exponent y crosses over to values of up to 2 when the intrinsic persistence length exceeds the electrostatic one . We find a pronounced minimum of with increasing intrinsic stiffness due to a reduction of entropy. The concept of a unique persistence length is not applicable for these systems.