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).

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