An analytical model for hole boring proton acceleration by a circularly-polarized CO₂ laser pulse in a gas jet is developed. The plasma density profile near the density peak is taken to be rectangular, with inner region thickness l around a laser wavelength and density 10% above the critical, while the outside density is 10% below the critical. On the rear side, plasma density falls off rapidly to a small value. The laser suffers strong reflection from the central region and, at normalized amplitude a₀≥1, creates a double layer. The space charge field of the double layer, moving with velocity v_fz^, reflects up-stream protons to 2v_f velocity, incurring momentum loss at a rate comparable to radiation pressure. Reflection occurs for v_f≤ω_p (z_flm/m_p)^-1/2 , where m and m_p are the electron and proton masses, z_f is the distance traveled by the compressed electron layer and ω_p is the plasma frequency. For Gaussian temporal profile of the laser and parabolic density profile of the upstream plasma, the proton energy distribution is narrowly peaked.