A process for selectively removing carbon dioxide from a gaseous stream by converting the carbon dioxide to a solid, stable form is provided. In a sequestration process, carbon dioxide enriched air is passed through a gas diffusion membrane to transfer the carbon dioxide to a fluid medium. The carbon dioxide rich fluid is then passed through a matrix containing a catalyst specific for carbon dioxide, which accelerates the conversion of the carbon dioxide to carbonic acid. In the final step, a mineral ion is added to the reaction so that a precipitate of carbonate salt is formed. This solid mineral precipitate can be safely stored for extended periods of time, such as by burying the precipitate in the ground or depositing the precipitate into storage sites either on land or into a body of water. An apparatus for removing carbon dioxide from a gaseous stream is also provided.