  Applying the quantum field theoretic perturbiner approach to Einstein
gravity, we compute the metric of a Schwarzschild black hole order by order in
perturbation theory. Using recursion, this calculation can be carried out in de
Donder gauge to all orders in Newton's constant. The result is a geometric
series which is convergent outside a disk of finite radius, and it agrees
within its region of convergence with the known de Donder gauge metric of a
Schwarzschild black hole. It thus provides a first all-order perturbative
computation in Einstein gravity with a matter source, and this series converges
to the known non-perturbative expression in the expected range of convergence.
