  We achieve fast, nondestructive quantum-state readout via fluorescence
detection of a single $^{87}$Rb atom in the 5$S_{1/2}$ ($F=2$) ground state
held in an optical dipole trap. The atom is driven by linearly-polarized
readout laser beams, making the scheme insensitive to the distribution of
atomic population in the magnetic sub-levels. We demonstrate a readout fidelity
of $97.6\pm0.2\%$ in a readout time of $160\pm20$ $\mu$s with the atom retained
in $>97\%$ of the trials, representing an advancement over other
magnetic-state-insensitive techniques. We demonstrate that the $F=2$ state is
partially protected from optical pumping by the distribution of the dipole
matrix elements for the various transitions and the AC-Stark shifts from the
optical trap. Our results are likely to find application in neutral-atom
quantum computing and simulation.
