Arms crossed, scowl set, President-elect Donald J. Trump avoided jail, but became a felon.

Mr. Trump appeared virtually at his criminal sentencing on Friday from his Mar-a-Lago estate in Florida, more than 1,000 miles away from the chilly Manhattan courtroom where his case was called for a final time. Projected on a 60-inch screen, his image loomed over the gallery as a prosecutor recounted his crimes and a judge imposed his sentence.

Mr. Trump once faced up to four years in prison for falsifying business records to cover up a sex scandal, but on Friday, he received only a so-called unconditional discharge. The sentence, a rare and lenient alternative to jail or probation, reflected the practical and constitutional impossibility of jailing a president-elect.

It nonetheless carried symbolic significance, capping a yearslong ordeal that consumed Mr. Trump as a weary nation reckoned with the prospect of a criminal president. Once the sentencing concluded, it cemented his status as the first felon to occupy the Oval Office.

“Never before has this court been presented with such a unique and remarkable set of circumstances,” said the trial judge, Juan M. Merchan, who has felt the brunt of Mr. Trump’s rage over the case the past two years. “This has been a truly extraordinary case.”

It was also “a bit of a paradox,” he said, as the somber ritual of sentencing testified to both the supremacy and limits of presidential power.

The fact the proceeding happened at all — despite Mr. Trump’s frenzied effort to shut it down — showed that the man preparing to reclaim the nation’s highest office was not entirely above the law. The machinery of New York’s justice system briefly brought him low, albeit in the comfort of his Florida mansion.

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