  The one-loop self-energy correction to the 1s electron g factor is evaluated
to all orders in Z\alpha with an accuracy, which is essentially better than
that of previous calculations of this correction. As a result, the uncertainty
of the theoretical prediction for the bound-electron g factor in H-like carbon
is reduced by a factor of 3. This improves the total accuracy of the recent
electron-mass determination [Beier et al. Phys. Rev. Lett. 88, 011603 (2002)].
The new value of the electron mass is found to be m_e = 0.000 548 579 909 3(3)
u.
