Llandeilo residents have been demanding action for more than 40 years.
It suffers from heavy congestion as lorries squeeze through the main street and air pollution levels have been found to exceed national standards.
The Welsh Government said construction should start by the end of 2019 and will take two years to complete.
Last week, it announced £50m funding for the route as part of £83m earmarked for transport in the budget.
The town's main street on the A483 is currently used as a route from Swansea to join the A40 to the Midlands.
Economy Secretary Ken Skates said: "We now expect to see shovels in the ground by the end of 2019 and anticipate the road being open two years later.
"Alongside our commitment to delivering the major improvements to the M4, A55, A40 and A494, I'm determined to ensure that our infrastructure improvements leave no-one behind."
A date has been set for work to start on a long-awaited bypass for a Carmarthenshire town which should ease congestion and reduce pollution.