ST. PAUL — Minnesota Gov. Mark Dayton and top Republicans agreed Thursday to end a budget impasse that prompted the longest state government shutdown in recent history, after the Democratic governor surrendered on raising taxes.

Dayton said the state government would be back in business “very soon,” but he didn’t say exactly when.

The deal to erase a $5 billion deficit came after a big sacrifice from Dayton, who made new income taxes a central campaign message last year and the centerpiece of his budget. He dropped that and said he would accept — with conditions — an offer the GOP put forward on the eve of the shutdown to bring about $1.4 billion into the budget by delaying payments to schools and selling tobacco payment bonds.

Republicans agreed to his conditions, which included relinquishing a list of policy changes such as banning state aid for stem cell research and a plan to cut the state workforce by 15 percent.

They conceded to higher state spending than they had wanted. Republican lawmakers spent months insisting that the two-year budget be capped at $34 billion, the amount the state was projected to collect without new sources of money. Instead, it will be closer to $35.4 billion.

The deal — if approved by lawmakers — would end a government interruption that has lasted two weeks and isn’t over yet.

Dayton announced the deal outside his office with House Speaker Kurt Zellers and Senate Majority Leader Amy Koch after a three-hour negotiating session. The somber looks on their faces testified to a hard bargain.

“It was about making sure that we get a deal that we can all be disappointed in, but a deal that is done, a budget that was balanced, a state that was back to work,” Zellers said.

“Nobody is going to be happy with this, which is the essence of real compromise,” Dayton said.

Dayton didn’t say when he will call a special legislative session to pass a budget but indicated it would be within days. He said a stopgap funding measure won’t be necessary because the two sides will agree on and pass bills setting a new two-year budget.

Yet many of the deal’s details remained murky, including exactly what will be cut from planned spending.

The shutdown has idled 22,000 state employees, closed state parks and rest stops and cut off funding to many social services. It has cost the state millions in the cost of preparing for the shutdown and in lost revenue since then. The interruption has also prevented entrepreneurs and professionals from getting state licenses. The latest licensing snag threatens to stop the sale of Miller, Coors and other popular beers in the state within days.

Payments by the state to schools and local governments have continued, and a court has taken some of the pressure off by restarting the flow of cash to programs ranging from child care assistance to home meal services for the elderly.

The governor sounded weary earlier Thursday when he told a University of Minnesota audience in Minneapolis that he would embrace the GOP proposal. He said people he met as he traveled around the state had this clear message: End the shutdown.

“They want this resolved, and they don’t even care how. I care how,” Dayton said.

The deal is contingent on approval by the Legislature, no easy task after an election in which a more conservative Republican caucus took power. Koch and Zellers said they believed rank-and-file legislators would approve it. Republicans hold narrow majorities in both chambers, and Democratic minority leaders weren’t in on the deal-making.

“Certainly we’re not doing any end zone dances,” said Rep. Mike Benson, a freshman Republican from Rochester. “Realistically there are some things that are going to go down hard. Sounds to me we’re kicking the can down the road a little bit with the education shift, but we’re not raising taxes.”

Democratic House Minority Leader Paul Thissen said in a statement that it would be up to Republicans to pass the compromise.

Unions, and some Democrats, sharply criticized the plan as irresponsible for borrowing against future revenue.

“More debt and more borrowing only make this bad situation worse,” state Rep. Ryan Winkler, a Democrat from Golden Valley, said in a statement. Winkler said the delay in school funding, which has become a regular part of Minnesota’s budget balancing, would “mortgage our children’s future.”

“Some of what we have been working for has been set back years today,” said Rev. Grant Stevensen, who heads a coalition of Twin Cities congregations focused on social justice issues and said he was disappointed that Dayton dropped his call to raise taxes on top earners.

Outside the Capitol, there was frustration that the impasse went far enough to close government.

“I guarantee I lost some business out of it,” said Jim Berg, who owns a 13-cabin resort in Crosslake and was hurt by the suspension of fishing licenses in the shutdown.

Berg said he’s not sure whether the settlement comes soon enough for him to salvage the rest of the summer.

“Only time’s going to tell that,” he said.