  A new local, covariant ``counter-term'' is used to construct a variational
principle for asymptotically flat spacetimes in any spacetime dimension $ d \ge
4$. The new counter-term makes direct contact with more familiar background
subtraction procedures, but is a local algebraic function of the boundary
metric and Ricci curvature. The corresponding action satisfies two important
properties required for a proper treatment of semi-classical issues and, in
particular, to connect with any dual non-gravitational description of
asymptotically flat space. These properties are that 1) the action is finite
on-shell and 2) asymptotically flat solutions are stationary points under {\it
all} variations preserving asymptotic flatness; i.e., not just under variations
of compact support. Our definition of asymptotic flatness is sufficiently
general to allow the magentic part of the Weyl tensor to be of the same order
as the electric part and thus, for d=4, to have non-vanishing NUT charge.
Definitive results are demonstrated when the boundary is either a cylindrical
or a hyperbolic (i.e., de Sitter space) representation of spacelike infinity
($i^0$), and partial results are provided for more general representations of
$i^0$. For the cylindrical or hyperbolic representations of $i^0$, similar
results are also shown to hold for both a counter-term proportional to the
square-root of the boundary Ricci scalar and for a more complicated
counter-term suggested previously by Kraus, Larsen, and Siebelink. Finally, we
show that such actions lead, via a straightforward computation, to conserved
quantities at spacelike infinity which agree with, but are more general than,
the usual (e.g., ADM) results.
