I took a countable dense set in R^n - C, x_j, j = 1, 2, ...,
and for each j took the closed circle with center x_j and radius
d(x_j, C) (distance from x_j to
C which is well defined due to
compactness).  Then I defined a
function g_j: R^n --&gt; R 0 outside the
circle and on the circle with
values on the radii much as in
the function Milnor, Rudin, and
I mentioned.  So, g_j is 
infinitely differentiable,
strictly positive on the interior
of the circle at x_j, and
0 otherwise.  Then I added
the functions g_j in a 
convergent way, again using
the exponential trick.
The resulting function 
was as desired, 0 on C,
strictly positive otherwise,
and infinitely differentiable.