  In measurements of small signals using spin precession the precession angle
usually grows linearly in time. We show that non-linear interactions between
particles can lead to an exponentially growing spin precession angle, resulting
in an amplification of small signals and raising them above the noise level of
a detection system. We demonstrate amplification by a factor of greater than 8
of a spin precession signal due to a small magnetic field gradient in a
spherical cell filled with hyperpolarized liquid $^{129}$Xe. This technique can
improve the sensitivity in many measurements that are limited by the noise of
the detection system, rather then the fundamental spin-projection noise.
