The production of a sterile species via active-sterile mixing in a thermal medium is studied in an exactly solvable model. The exact time evolution of the sterile distribution function is determined by the dispersion relations and damping rates ${\ensuremath{\Gamma}}_{1,2}$ for the quasiparticle modes. These depend on $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}={\ensuremath{\Gamma}}_{aa}/2\ensuremath{\Delta}E$, with ${\ensuremath{\Gamma}}_{aa}$ the interaction rate of the active species in absence of mixing and $\ensuremath{\Delta}E$ the oscillation frequency in the medium without damping. $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}\ensuremath{\ll}1$, $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}\ensuremath{\gg}1$ describe the weak and strong damping limits, respectively. For $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}\ensuremath{\ll}1$, ${\ensuremath{\Gamma}}_{1}={\ensuremath{\Gamma}}_{aa}{cos}^{2}{\ensuremath{\theta}}_{m}$; ${\ensuremath{\Gamma}}_{2}={\ensuremath{\Gamma}}_{aa}{sin}^{2}{\ensuremath{\theta}}_{m}$ where ${\ensuremath{\theta}}_{m}$ is the mixing angle in the medium and the sterile distribution function does not obey a simple rate equation. For $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}\ensuremath{\gg}1$, ${\ensuremath{\Gamma}}_{1}={\ensuremath{\Gamma}}_{aa}$ and ${\ensuremath{\Gamma}}_{2}={\ensuremath{\Gamma}}_{aa}{sin}^{2}2{\ensuremath{\theta}}_{m}/4{\stackrel{\texttildelow{}}{\ensuremath{\gamma}}}^{2}$, is the sterile production rate. In this regime sterile production is suppressed and the oscillation frequency vanishes at an Mikheyev-Smirnov-Wolfenstein (MSW) resonance, with a breakdown of adiabaticity. These are consequences of quantum Zeno suppression. For active neutrinos with standard model interactions the strong damping limit is only available near an MSW resonance if $\mathrm{sin}2\ensuremath{\theta}\ensuremath{\ll}{\ensuremath{\alpha}}_{w}$ with $\ensuremath{\theta}$ the vacuum mixing angle. The full set of quantum kinetic equations for sterile production for arbitrary $\stackrel{\texttildelow{}}{\ensuremath{\gamma}}$ are obtained from the quantum master equation. Cosmological resonant sterile neutrino production is quantum Zeno suppressed relieving potential uncertainties associated with the QCD phase transition.