We report observations of the ionization state of anomalous cosmic-ray (ACR) nitrogen, oxygen, and neon during the period 1992 October to 1993 May, carried out with instrumentation on the Solar, Anomalous & Magnetospheric Particle Explorer (SAMPEX) spacecraft. The low-altitude (510 x 675 km) and high-inclination (82 deg) orbit enables SAMPEX to sample the interplanetary ACR fluxes on each polar pass and then to observe the cutoff of these fluxes by the geomagnetic field at lower latitudes. The arrival time and direction of each ion is recorded by the instruments, allowing detailed calculations of the particle's trajectory through the Earth's magnetic field and thereby placing upper limits on the ionization state of the particles. We find (a) that ACR nitrogen, oxygen, and neon each contain singly ionized particles and (b) that ACR oxygen is predominantly singly ionized with an upper limit of 10% for higher ionization states. These ionization states confirm theories of ACR origin as neutral interstellar material that is singly ionized near the Sun by UV or charge exchange with the solar wind, and is subsequently accelerated in the outer heliosphere.