Summary: We study a three-point mapping that has the Painlevé-II \(\text{P}_{\text{II}}\) equation as continuous limit. The qualifier ‘alternate’ is due to the fact that there already exists another discrete form of \(\text{P}_{\text{II}}\). We present the Lax pair for this discrete equation, as well as its auto-Bäcklund transform. Special rational solutions as well as solutions in terms of the (discrete) Airy function are obtained for particular values of the parameter of the equation.