The burglars "severely damaged" the ancient urn that contained the couple's remains when they attempted to take it from Golders Green Crematorium, north west London between New Year's Eve and New Year's Day, police said.
Detective constable Daniel Candler called the attempted theft "despicable" as he appealed for information.
He said: "This was a despicable act by a callous thief. Even leaving aside the financial value of the irreplaceable urn and the historical significance of to whom it related; the fact that someone set out to take an object knowing it contained the last remains of a person defies belief. I urge anyone who knows the perpetrator to contact me or Crimestoppers."
The urn dates back to the third century BC.
Anyone with information can call Mr Candler on 0208 733 4525 or Crimestoppers anonymously on 0800 555 111.