President Trump’s promise to investigate voter fraud has drawn predictable responses from Democrats and the media, who insist there is no such thing and have been fighting for years to prevent any inquiry into the matter. But an investigation in Mr. Trump’s hometown shows that the problem is real.

In 2013 the New York City Department of Investigation—the storied law-enforcement arm of city government, which houses and manages all the city’s inspectors general and investigators—decided to test the system. City investigators posed as 63 ineligible individuals still on the city voter rolls. Each ineligible voter had died, moved out of the jurisdiction, or been convicted of a felony at least two years earlier.

The investigators didn’t go to great lengths to hide their attempted fraudulent votes. In five instances investigators in their 20s or 30s posed as voters age 82 to 94. In some cases the investigators were of different ethnic backgrounds from the voters they were impersonating. Yet each was given a ballot and allowed to cast a vote without question.

In other instances the investigators informed the poll worker that they had moved but didn’t have time to get to their new home on Election Day; all but one was allowed to vote. Only one investigator was flat-out rejected. He had the misfortune of trying to vote at a polling place where the clerk was the mother of the ineligible felon he was impersonating.

Ninety-seven percent of the barely disguised phony voters were allowed to vote unimpeded, and none was referred for criminal charges or officially reported to the Board of Elections. One can only imagine what a sharp operator trying to fix an election could do by flooding polling places with ineligible voters.

The Department of Investigation proved how easy it is to perpetrate such a fraud. More important, it showed that such illegal behavior doesn’t get reported or corrected. When the department published its report, the Board of Elections took no steps to prevent future fraud. Instead, the board complained that the investigators engaged in voter fraud and should be prosecuted. The district attorney refused to charge the investigators for their productive and lawful investigation.

Last year a Democratic New York election official, Alan Schulkin, was forced to resign after he was caught on tape telling the truth about electoral process. “Certain neighborhoods in particular, they bus people around to vote,” Mr. Schulkin said. “They put them in a bus and go poll site to poll site.” In another case a local polling official reported to the police that a group of men had apparently voted more than once, while others were told they couldn’t vote because their ballots had already been cast. The police response, according to press reports, was that they don’t investigate alleged voter fraud.

In 2014 President Obama’s bipartisan Presidential Commission on Election Administration concluded that maintaining accurate election rolls was a key part of the democratic process. Yet subsequent efforts to do so has led to claims of voter suppression, thereby stalling progress.

And while New York does not require identification to vote, its registration requirements are fairly strict and there is no such thing as same-day registration. Many states are less stringent, which would make the opportunities for fraud greater.

America has a flawed voting system that is susceptible to fraud and has few precautions against it. A federal investigation could go a long way toward ensuring citizens U.S. elections are honest.

Mr. Levy is vice chairman of the Republican National Lawyers Association.