  Several next-generation experiments aim to make the first measurement of the
neutrino flux from the Carbon-Nitrogen-Oxygen (CNO) solar fusion cycle. We
calculate how much time these experiments will need to run for in order to
measure this flux with enough precision to tell us the metal content of the
Sun's core, and thereby help to solve the solar metallicity problem. For
experiments looking at neutrino-electron scattering, we find that SNO+ will
measure this CNO neutrino flux with enough precision after five years in its
pure scintillator mode, provided its $^{210}$Bi background is measured to 1%
accuracy. By comparison, a 100~ton liquid argon experiment such as Argo will
take ten years in Gran Sasso lab, or five years in SNOLAB or Jinping. Borexino
could obtain this precision in ten years, but this projection is very sensitive
to background assumptions. For experiments looking at neutrino-nucleus
scattering, the best prospects are obtained for low-threshold solid state
detectors (employing either germanium or silicon). These would require new
technologies to lower the experimental threshold close to detection of single
electron-hole pairs, and exposures beyond those projected for next-generation
dark matter detectors.
