We have studied the kinetics of the reaction of a proton with a mild base in ice. The proton was injected to the ice crystal by a proton-transfer reaction from an electronically excited photoacid. Mild bases react with the transferred proton with a large intrinsic rate constant. We used our diffusion assisted geminate recombination model based on the Debye-Smolochowski equation with an additional term to account for the proton scavenging by the base. We find that in liquid water the measured scavenging effect is slightly smaller than expected from a diffusion controlled reaction rate constant. In ice, the scavenging effect is rather large, and the diffusion controlled reaction rate constant underestimates the measured proton scavenging effect in a wide range of concentrations and temperature. We explain the large effective proton scavenging in ice by the tendency of ice formation to concentrate the impurities to confined volumes in order to minimize the ice crystal energy.