(My expectation is that the proof has to be replicated and the
conviction cannot be introduced as evidence).
Your expectation is incorrect.
The name of the legal doctrine that allows a criminal judgment to have this effect in a civil case is called "collateral estoppel" which is also sometimes called "issue preclusion". See, e.g., A-1 Auto Repair & Detail, Inc. v. Bilunas-Hardy, 93 P.3d 598, 600 (Colo. App. 2004) ("Hardy contends Colorado law does not allow courts to apply collateral estoppel, now commonly known as the doctrine of issue preclusion, when the first adjudication is criminal and the subsequent litigation is civil. We disagree.")
Similarly, a case out of California stated:
To preclude a civil litigant from relitigating an issue previously
found against him in a criminal prosecution is less severe than to
preclude him from relitigating such an issue in successive civil
trials, for there are rigorous safeguards against unjust conviction,
including the requirements of proof beyond a reasonable doubt and of a
unanimous verdict, the right to counsel, and a record paid for by the
state on appeal. Stability of judgments and expeditious trials are
served and no injustice done, when criminal defendants are estopped
from relitigating issues determined in conformity with these
Teitelbaum Furs, Inc. v. Dominion Ins. Co., 58 Cal.2d 601, 606, 25 Cal.Rptr. 559, 375 P.2d 439, 441 (1962) (citations omitted).
To the best of my knowledge, this is the rule in every U.S. jurisdiction (with the possible exceptions of Puerto Rico and Louisiana which are not common law jurisdictions). It is also the historical rule in British common law, although I don't know if this continues to be the case in non-U.S. jurisdictions.
Procedurally, the determination that collateral estoppel applies would usually be made on a motion for summary judgment, or in the preparation of jury instructions which state that liability has been established and that the jury is to limit itself to determining causation and damages, rather than as an evidentiary matter.
I've used this doctrine once or twice.
For example, I used it in a case where someone fraudulently sold ditch company shares worth several hundred thousand dollars (in Colorado, water is gold) that he didn't own (a transaction that could not be unwound because the buyer was a bona fide purchaser for value and the seller had apparent authority as a trustee of a trust owning the shares even though he didn't have the actual authority to sell them under the trust) and then spent the money he received before he was discovered (if I recall correctly, for gambling debts). He was convicted criminally and then my client, the victim, sued for money damages including statutory treble damages for civil theft and attorney's fees based upon collateral estoppel and an affidavit as to damages in a motion for summary judgment.
From a practical perspective the four main difficulties are that (1) people convicted of crimes often lack the income or assets to pay judgments, (2) there are double recovery issues involved in reconciling restitution awards in a criminal case (where the measure of damages is narrower) and damage awards in a civil case (where the measure of damages is broader), (3) there are priority issues involved in reconciling criminal awards for fines, restitution and costs, in each case with civil awards for damages, and (4) if the defendant declares bankruptcy, the non-dischargeability of the civil judgment must be affirmatively raised and proved (often this is elementary but there are strict time limits) in the bankruptcy proceeding.
Tactically, it is often better to sue first, collect what you can, and to bring a criminal complaint only when it turns out that the perpetrator is judgment-proof.