In a court of law, what is proof positive of a prior felony conviction?

?? That is a matter of record, the judge can confirm that any time he chooses. A simple printout of the record will suffice.

In most cases, prior record is only a factor in sentencing, so the prosecutor will not normally bring it into evidence.

A certified copy of the conviction or case disposition from the court clerk of the jurisdiction where they were convicted.

EDIT: But unless the defendant denied it when they were being cross examined and the prosecutor needed it to impeach the defendant, then I guess having a copy of it wouldn't be necessary. This is because if the defendant never takes the stand, or more specifically never puts their character in issue (or calls a witness who puts their character at issue), the state can't just bring it up during trial. The defendant has to open the door either by his own doings or through a witness that the defense calls. With all that said, however, during sentencing, rules of evidence don't apply, so the prosecutor could bring it to the attention of the judge so that the judge can make their sentence accordingly.

There are exceptions however. If the defendant's criminal conviction goes towards proving something other than propensity, then they can use it without the defendant or their witnesses ever bringing an issue of character up. So if it somehow is used to prove identity, knowledge, plan, opportunity, etc., and not a propensity to commit crimes, then they can use it. Another exception is if the prior crime was a crime of dishonesty since it would be relevant to their credibility for telling the truth on the stand. But other than those exceptions, the defendant must first open the door. And yea, like other posters said, since most courts are all online now, they have pretty easy access to not only their state or jurisdiction's criminal data bases, but also they can easily access other jurisdiction's data bases. Before all the going online stuff, if the conviction was from another state, then they would probably have had to get it sent over to them by the other court.

If there is an issue, merely the defendant's criminal FBI record, but the court would already have a copy of that. Criminal histories are located inside the court case packet.