The contractors behind the AWPR are set to receive an additional £65 million.
The Scottish Government is due to make the payments despite the deal for the bypass involving a fixed price of £775 million.
However, contractors claimed delays had pushed the cost of the project above £1 billion – and were prepared to take the issue to court.
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Although ministers insist they were right, they have decided to pay out in order to avoid a legal battle.
In a letter to Edward Mountain MSP, the convener of the rural economy and connectivity committee, transport secretary Michael Matheson said: “Whilst the Scottish Ministers were prepared to mount a robust defence, it was recognised that proceedings could have run for years and led to significant and prolonged exposure to risk, expense and uncertainty as a result.
“Ultimately, all parties appreciated that an early commercial settlement was desirable in all the circumstances.”