  Interest has focussed recently on low energy implications of a nontrivial
scale invariant sector of an effective field theory with an IR fixed point,
manifest in terms of ``unparticles'' with peculiar properties. If unparticle
stuff exists it could couple to the stress tensor and mediate a new 'fifth'
force which we call 'ungravity' arising from the exchange of unparticles
between massive particles, which in turn could modify the inverse square law.
Under the assumption of strict conformal invariance in the hidden sector down
to low energies, we compute the lowest order ungravity correction to the
Newtonian gravitational potential and find scale invariant power law
corrections of type $(R_{G}/r)^{2d_{\cal U} -1}$ where $d_{\cal U}$ is an
anomalous unparticle dimension and $R_{G}$ is a characteristic length scale
where the ungravity interactions become significant. $d_{\cal U}$ is
constrained to lie the range $ d_{\cal U} > 3 (2)$ for a spin 2 (spin 0)
unparticle coupling to the stress tensor (and its trace) and leads to
modification of the inverse square law with $r$ dependence in the range between
$1/r^{4+2\delta} (\delta>0)$, while extra dimension models with warping modify
the force law with corrections beginning with terms O$(1/r^3)$ for small $r$
but exponentially suppressed for large $r$. Thus a discrimination between extra
dimension models and ungravity is possible in future improved submillimeter
tests of gravity.
