Without killing the mold, it will continue to pose a serious health risk to you. First, you must figure out what caused the mold in the first place and rectify that problem. The most common reason aside from surface wetting is lack of air flow under the mattress and not turning it over frequently.
As to removing the mold, I would recommend steam. The most economical steamer suited to the job would be the SteamFast SF275.
Steam kills mold, germs, and bacteria on contact without the use of toxic or caustic chemicals and can be used for all kinds of cleaning and sanitizing tasks around the home.