Amendments are strange critters ~ most of them that are actually proposed on the floor of either House die right there. Most of those voted for by one house die in the other. Most of those voted for by both houses die in the states. Even at the end, if they are approved by enough states, they require some sort of certification as having been passed back in DC.
They have a rough road.
The balanced budget amendment could well be passed within hours and be on its way to the states ~ and it will never be seen again.