Theoretical studies are presented how the electric dipole moment (EDM) of the electron in H-like ions in electrostatic storage rings can sensitively be determined. With the proposed experiments a new constraint of about $10^{-29}$ e cm for the electron EDM can be established what is by an order of magnitude more restrictive than the existing bounds. Experiments with H-like ions may provide a possibility to distinguish between the electron EDM effect and the effect of P,T violating interaction between the atomic electron and the nucleus.