Mayor Rahm Emanuel is off to Rome to witness the elevation of Archbishop Blase Cupich to cardinal after the City Council on Wednesday approved his $8.2 billion budget in record time without a single dissenting vote.

But now comes the hard part: Delivering on the mayor’s $60 million first-year promise to fill 471 police vacancies, keep pace with rising retirements and still hire enough police officers in 2017 to add 250 patrol officers, 37 sergeants, 50 lieutenants, 92 field-training officers and 100 detectives.

Former Mayor Richard M. Daley once promised to bolster the police force by 600 officers, only to “run out of money,” according to then-Police Supt. LeRoy Martin.

Martin’s candid admission to aldermen so embarrassed and infuriated Daley, the police superintendent abruptly retired a few months later.

The question is whether Emanuel can avoid falling into the same trap.

The mayor boldly predicted that he can and he will.

He noted that “one of the biggest classes” in Chicago history recently graduated from a police academy that will be converted into a factory for recruits, with training of veteran officers and candidates for promotion being shifted to City Colleges and DeVry University.

Chicago will also hold another police entrance exam in April and reduce the “pre-employment process” by up to two months.

“We have the resources to do it, and we’re going to do it. . . . The very things that you have to do, kind of operationally, to get to the goals that I’ve set we’re actually already putting in place,” the mayor said.

“We’re on record for a significant year this year, keeping pace with what we need to do with the force and also doing what we need to do operationally to achieve that goal — from test giving to freeing up space in the training facility for more people to be trained and ready to hit the streets,” he said.

National Fraternal Order of Police Executive Director Jim Pasco has questioned Emanuel’s ability to add 529 police officers in 2017 in a Chicago Police Department still under a sweeping federal civil rights investigation triggered by the police shooting of Laquan McDonald.

The police hiring surge marked a stunning turnaround for a mayor who closed three police stations; eliminated more than 1,400 police vacancies; and spent five years relying on overtime — to the tune of $116.1 million a year — to mask a manpower shortage that has mushroomed under Emanuel with police retirements outpacing hiring by 975 officers.

“It would be extraordinarily difficult under any circumstances, and particularly difficult, given the fact that so much has been done to besmirch the reputation of the Chicago Police Department,” Pasco said.

“This constant drumbeat of media and political criticism of the effectiveness of the department and the conduct of some officers has had a deleterious effect on the perception of the department,” he said. “When that happens, it makes it more difficult to recruit and train qualified applicants.”

Chicago aldermen are equally skeptical.

“I don’t think he can do it year one. They’re gonna have a hard time really pulling that together. It could be a wash with the amount of people who retire within a year,” said Ald. Scott Waguespack (32nd).

“Our concern is kind of moving the chips around and making sure that they all add up. And that hasn’t proven to be true over the last few years.”

Even Ald. Pat O’Connor (40th), the mayor’s City Council floor leader, was not so sure Emanuel can deliver on the ambitious promise to hire 529 police officers over and above attrition.

“We are going to give it every effort. We’re gonna try and keep up with the schedule that we’ve put out there. And if it’s not something we can meet, we’ll continue to try and add as many as we can,” O’Connor said.

Whether or not Emanuel delivers on the police hiring promise, Wednesday’s unanimous vote was a great sendoff for his pre-Thanksgiving trip.

Emanuel and an 80-member delegation that includes a handful of aldermen and city officials are off to Rome — and they acted like they had a plane to catch. The budget debate was the shortest in recent memory, and it was no debate at all.

The only aldermen to speak were Finance Committee Chairman Edward Burke (14th) and Ald. Jason Ervin (28th), who chaired two weeks of budget hearings for an ailing Budget Committee Chairman Carrie Austin.

Both sang the praises of the mayor’s budget.

After the 48-0 vote, Emanuel applauded aldermen for the three massive tax increases for pensions they approved over the last two years, setting the stage for Wednesday’s vote to be so easy.

“Yes, it was kind of quick. Yes, it was kind of quiet. But what preceded it was a lot of what you would want out of a public debate over five years,” Emanuel said.

“You have to think about this over five years [and] where we were in 2011. A lot of people thought being able to tackle Chicago’s fiscal challenges was almost a task too big for the political will and capacity of the city. So, while it may have been silent at this moment, what I take from today is we’re up to the challenge.”

While the budget vote was unanimous, the roll call on the $50 million revenue package that includes a 7-cents-a-bag tax on disposable bags was not.

“No” votes were cast by Aldermen Ricardo Munoz (22nd), Scott Waguespack (32nd) and Carlos Ramirez-Rosa (35th).

All three no votes were tied to Emanuel’s plan to create a $100 million Catalyst Fund to bankroll neighborhood projects.

Munoz, Waguespack and Ramirez-Rosa all wanted assurances they didn’t get that the money would be spent in impoverished neighborhoods.

They’re also opposed to the fact that, like Emanuel’s slow-starting Infrastructure Trust, the Catalyst Fund will be not be under the watchful eye of Inspector General Joe Ferguson, nor will it be required to follow Freedom of Information laws.