  Unparticles as suggested by Georgi are identities that are not constrained by
dispersion relations but are governed by their scaling dimension, d. Their
coupling to particles can result in macroscopic interactions between matter,
that are generally an inverse nonintegral power of distance. This is totally
different from known macroscopic forces. We use the precisely measured
long-ranged spin-spin interaction of electrons to constrain unparticle
couplings to the electron. For 1<d<1.5 the axial vector unparticle coupling is
excluded; and for 1<d<1.3 the pseudoscalar and vector couplings are also ruled
out. These bounds and the ones for other ranges of d exceed or are
complementary to those obtained previously from exotic positronium decays.
