Crematorium

Since you asked about ashes OR remains, I think the word you're looking for is "mausoleum". It could hold either. A building to hold cremated remains only is a "columbarium". A "crematorium" is the building where the actual cremation takes place.

Probably the most commonly used word is a "mausoleum" but a mausoleum can contain coffins with uncremated remains and/or urns with the cremated ashes of the deceased. http://www.ehow.com/facts_5817372_definition-mausoleum.html The technically correct word is "columbarium" but that is not as widely known as "mausoleum". http://www.ehow.com/about_5070959_columbarium.html