  Dynamical instabilities due to spin-mixing collisions in a $^{87}$Rb F=1
spinor Bose-Einstein condensate are used as an amplifier of quantum spin
fluctuations. We demonstrate the spectrum of this amplifier to be tunable, in
quantitative agreement with mean-field calculations. We quantify the
microscopic spin fluctuations of the initially paramagnetic condensate by
applying this amplifier and measuring the resulting macroscopic magnetization.
The magnitude of these fluctuations is consistent with predictions of a
beyond-mean-field theory. The spinor-condensate-based spin amplifier is thus
shown to be nearly quantum-limited at a gain as high as 30 dB.
