  We experimentally demonstrate stable trapping of a permanent magnet sphere
above a lead superconductor, in vacuum pressures of $4 \times 10^{-8}$~mbar.
The levitating magnet behaves as a harmonic oscillator, with frequencies in the
4-31~Hz range detected, and shows promise to be an ultrasensitive acceleration
sensor. We directly apply an acceleration to the magnet with a current carrying
wire, which we use to measure a background noise of $\sim 10^{-10} \
\text{m}/\sqrt{\text{Hz}}$ at 30.75~Hz frequency. With current experimental
parameters, we find an acceleration sensitivity of $S_a^{1/2} = 1.2 \pm 0.2
\times 10^{-10} \ \text{g}/\sqrt{\text{Hz}}$, for a thermal noise limited
system. By considering a 300~mK environment, at a background helium pressure of
$1 \times 10^{-10}$~mbar, acceleration sensitivities of $S_a^{1/2} \sim 3
\times 10^{-15} \ \text{g}/\sqrt{\text{Hz}}$ could be possible with ideal
conditions and vibration isolation. To feasibly measure with such a
sensitivity, feedback cooling must be implemented.
