'A date which will live in infamy'

It's a Pearl Harbor Day tradition for me to reflect on FDR's word choice, but I feel dirtier and dirtier about it. Yeah, "that" is preferable with the restrictive clause. But "which" is perfectly comprehensible, the distinction is more of a custom than a rule, and anyway it's silly to complain about a timeless phrase. Of the seven words in that phrase, "which" is the least important.

More interesting to me is the popularity of the tuneless variant "a day that shall live in infamy." Would you believe that returns a third as many Google results as the real thing?