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.
This is not true. Many states have what is called a corpus delicti rule, which requires that a confession be corroborated by other evidence in order to sustain a conviction. I am inclined to assert that this is actually the law in most US states, but I haven't researched it.
IANAL, but I thought corpus delicti only means there needs to be evidence that the crime has been committed, not that the crime has been committed by the person confessing to it.
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.