I'm not sure they would stop growing this early in the season, though.
If you think about it, when the tops are frosted, they regrow from lower down the stem, don't they?
Potatoes stored below ground, in clamps' like people used to in the 'old days', were always slugged and some rotten, if we got any wet weather.
If you really can't store them above ground, you may as well try cutting the tops off and see what happens. You've nothing to lose, have you? If you do please let us know if it works