We achieve fast, nondestructive quantum-state readout via fluorescence detection of a single Rb87 atom in the 5S1/2 (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 magnetic sublevels. We demonstrate a readout fidelity of 97.6±0.2% in a readout time of 160±20µs 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.