  The star formation rate (SFR) is one of the most fundamental parameters of
galaxies, but nearly all of the standard SFR diagnostics are difficult to
measure in active galaxies because of contamination from the active galactic
nucleus (AGN). Being less sensitive to dust extinction, the mid-infrared
fine-structure lines of [NeII] 12.81 micron and [NeIII] 15.56 micron
effectively trace the SFR in star-forming galaxies. These lines also have the
potential to serve as a reliable SFR indicator in active galaxies, provided
that their contribution from the AGN narrow-line region can be removed. We use
a new set of photoionization calculations with realistic AGN spectral energy
distributions and input assumptions to constrain the magnitude of [NeII] and
[NeIII] produced by the narrow-line region for a given strength of [NeV] 14.32
micron. We demonstrate that AGNs emit a relatively restricted range of
[NeII]/[NeV] and [NeIII]/[NeV] ratios. Hence, once [NeV] is measured, the AGN
contribution to the low-ionization Ne lines can be estimated, and the SFR can
be determined from the strength of [NeII] and [NeIII]. We find that AGN host
galaxies have similar properties as compact extragalactic HII regions, which
indicates that the star formation in AGN hosts is spatially concentrated. This
suggests a close relationship between black hole accretion and nuclear star
formation. We update the calibration of [NeII] and [NeIII] strength as a SFR
indicator, explicitly considering the effects of metallicity, finding very good
relations between Ne fractional abundances and the [NeIII]/[NeII] ratio for
different metallicities, ionization parameters, and starburst ages. Comparison
of neon-based SFRs with independent SFRs for active and star-forming galaxies
shows excellent consistency with small scatter ($\sim0.18$ dex).
