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We present -- and make publicly available -- accurate and precise photometric redshifts in the ACS footprint from the COSMOS field for objects with $i_{\mathrm{AB}}\leq 23$. The redshifts are computed using a combination of narrow band photometry from PAUS, a survey with 40 narrow bands spaced at $100Å$ intervals covering the range from $4500Å$ to $8500Å$, and 26 broad, intermediate, and narrow bands covering the UV, visible and near infrared spectrum from the COSMOS2015 catalogue. We introduce a new method that models the spectral energy distributions (SEDs) as a linear combination of continuum and emission line templates and computes its Bayes evidence, integrating over the linear combinations. The correlation between the UV luminosity and the OII line is measured using the 66 available bands with the zCOSMOS spectroscopic sample, and used as a prior which constrains the relative flux between continuum and emission line templates. The flux ratios between the OII line and $\mathrm{H}_α$, $\mathrm{H}_β$ and $\mathrm{OIII}$ are similarly measured and used to generate the emission line templates. Comparing to public spectroscopic surveys via the quantity $Δ_z\equiv(z_{\mathrm{photo}}-z_{\mathrm{spec}})/(1+z_{\mathrm{spec}})$, we find the photometric redshifts to be more precise than previous estimates, with $σ_{68}(Δ_z) \approx (0.003, 0.009)$ for galaxies at magnitude $i_{\mathrm{AB}}\sim18$ and $i_{\mathrm{AB}}\sim23$, respectively, which is $3\times$ and $1.66\times$ tighter than COSMOS2015. Additionally, we find the redshifts to be very accurate on average, yielding a median of the $Δ_z$ distribution compatible with $|\mathrm{median}(Δ_z)|\leq0.001$ at all redshifts and magnitudes considered. Both the added PAUS data and new methodology contribute significantly to the improved results.