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Interest has focussed recently on low energy implications of a nontrivialscale invariant sector of an effective field theory with an IR fixed point,manifest in terms of ``unparticles'' with peculiar properties. If unparticlestuff exists it could couple to the stress tensor and mediate a new 'fifth'force which we call 'ungravity' arising from the exchange of unparticlesbetween massive particles, which in turn could modify the inverse square law.Under the assumption of strict conformal invariance in the hidden sector downto low energies, we compute the lowest order ungravity correction to theNewtonian gravitational potential and find scale invariant power lawcorrections of type $(R_{G}/r)^{2d_{\cal U} -1}$ where $d_{\cal U}$ is ananomalous unparticle dimension and $R_{G}$ is a characteristic length scalewhere the ungravity interactions become significant. $d_{\cal U}$ isconstrained 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 tomodification 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 modifythe force law with corrections beginning with terms O$(1/r^3)$ for small $r$but exponentially suppressed for large $r$. Thus a discrimination between extradimension models and ungravity is possible in future improved submillimetertests of gravity.

Scalar modifications to gravity from unparticle effects may be testable