In the US, at least, a confession is always considered sufficient evidence. It probably shouldn't be, since interrogators have managed to get people to confess to crimes that evidence was later found to exonerate.
So if they find a body with multiple stab wounds and I go in and say I stabbed the person, I can be convicted. If I say I murdered Audrey Farber, and the police can't even find a record of an Audrey Farber gone missing, then I can't be convicted because there's no evidence than an Audrey Farber was murdered at all.