We've got a stone birdbath where the bottom is unsealed, either from manufacture or age. The bowl of it looks like this.
The algal growth has gone into the stone and is impossible to remove with cleansers or bleach. Looking it up on the web, it seems as though the best way to deal with this is to sandblast it clean and then give it some kind of epoxy coating like you'd use on a garage floor.
Updates as events warrant.