The proposed new road will divert traffic away from the landslip-prone Kex Gill, mid-way between Skipton and Harrogate.
Where it runs through Blubberhouses Moor, the A59 is subject to regular landslips, closing the road, most recently in 2018.
In 2019 North Yorkshire County Council submitted proposals to the Department for Transport for a permanent solution, which involved realigning the route to the other side of the valley.
DfT has agreed to stump up £56m for the scheme; the council will provide the other £4m that is estimated to be needed.
The A59 is one of three schemes approved for DfT funding today. The others are: a £24m junction improvement in Sandwell to reduce congestion at Birchley Island and the intersection of the M5, A4123 and A4034; and £13m worth of upgrades to the Redbridge Causeway bridges over the River Test near Southampton.