The cross section for photon production in association with at least one jet containing a $b$-quark hadron has been measured in proton antiproton collisions at $\sqrt{s}=1.96$ TeV. The analysis uses a data sample corresponding to an integrated luminosity of 340 pb$^{-1}$ collected with the CDF II detector. Both the differential cross section as a function of photon transverse energy $E_T^{\gamma}$, $d \sigma$($p \overline{p} \to \gamma + \geq 1 b$-jet)/$d E_T^{\gamma}$ and the total cross sectio