The measured properties of the epoch of reionization (EoR) show that reionization probably began around $z∼ 12-15$ and ended by $z=6$. In addition, a careful analysis of the fluctuations in the cosmic microwave background indicate a scattering optical depth $τ∼ 0.066±0.012$ through the EoR. In the context of $Λ$CDM, galaxies at intermediate redshifts and dwarf galaxies at higher redshifts now appear to be the principal sources of UV ionizing radiation, but only for an inferred (ionizing) escape fraction $f_{ion}∼ 0.2$, which is in tension with other observations that suggest a value as small as $∼ 0.05$. In this paper, we examine how reionization might have progressed in the alternative Friedmann-Robertson Walker cosmology known as the $R_{\rm h}=ct$ Universe, and determine the value of $f_{ion}$ required with this different rate of expansion. We find that $R_{\rm h}=ct$ accounts quite well for the currently known properties of the EoR, as long as its fractional baryon density falls within the reasonable range $0.026\lesssim Ω_b\lesssim 0.037$. This model can also fit the EoR data with $f_{ion}∼ 0.05$, but only if the Lyman continuum photon production is highly efficient and $Ω_b ∼ 0.037$. These results are still preliminary, however, given their reliance on a particular form of the star-formation rate density, which is still uncertain at very high redshifts. It will also be helpful to reconsider the EoR in $R_{\rm h}=ct$ when complete structure formation models become available.