  We derive an action whose equations of motion contain the Poisson equation of
Newtonian gravity. The construction requires a new notion of Newton--Cartan
geometry based on an underlying symmetry algebra that differs from the usual
Bargmann algebra. This geometry naturally arises in a covariant $1/c$ expansion
of general relativity with $c$ the speed of light. By truncating this expansion
at subleading order we obtain the field content and transformation rules of the
fields that appear in the action of Newtonian gravity. The equations of motion
generalize Newtonian gravity by allowing for the effect of gravitational time
dilation due to strong gravitational fields.
