  We include effects of self-gravitation in the self-interaction of single
electrons with the electromagnetic field. When the effect of gravitation is
included there is an inevitable cut-off of the k-vector - the upper limit is
finite. The inward pressure of the self-gravitating field balances the outward
pressure of self-interaction. Both pressures are generated by self-interactions
of the electron with two fields - the vacuum electromagnetic field and the
self-induced gravitational field. Specifically we demonstrate that
gravitational effects must be included to stabilize the electron. We use the
Einstein equation to perform an exact calculation of the bare mass and electron
radius. We find a close-form solution. We find the electron radius $r_{e}
=9.2\sqrt{\alpha / 4\pi } \sqrt{\hbar G/c^{3}}$ $ =9.2\sqrt{\alpha / 4\pi
}l_{P}$ $ \approx 10^{ -36}m$ . $\sqrt{\hbar G/c^{3\text{}}}$ is the Planck
length $\ell _{P}$ , which is educed from first principles. We find that the
electromagnetic and gravitational fields merge at $\left (8/3\right
)\sqrt{\alpha / 4\pi } \sqrt{\hbar c/G}$ $ =\left (8/3\right )\sqrt{\alpha /
4\pi }\ m_{P} =10^{17} G e V$ in terms of the Planck mass $m_{P}$ .
Renormalisation is accomplished by requiring continuity of the interior and
exterior metrics at $r_{e}$ .
