I do the same as Rich, a week or so and then I cut them and freeze the meats.
The refrigerator is better than ice because the ice is not as consistent in keeping all of the clams the same temperature plus any clams on the bottom sitting in any fresh water will go bad due to the water.
Clams will open slightly to let air mix with the salt fluids inside to keep oxygen in the system while in a refrigerator but fresh water ice melt will kill clams.
I use stainless steel pans and place each clam in the pan on its side so when the shells open the fluids stay inside instead of draining out. A dry clam dies earlier than one kept on its side and moist.