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."

Relieved that the giant telecoms company would not be broken up, they piled into the shares, sending them up 3% in early trading.
BT dodged a bullet - and, as the chief executive of Ofcom, Sharon White, admitted, it was for prosaic reasons.
She said complications with land deals and BT's giant pension scheme meant there were "practical obstacles" to a break-up that would delay the process several years.
It's the pension scheme that probably most influenced Ofcom's thinking. BT's retirement scheme, inherited from its time as a state-owned utility, has assets of about Â£40bn and a deficit, on some measures, of about Â£10bn, even though BT has poured in billions of pounds in recent years to redress the gap.
Senior sources at the company say that about 80% of the scheme relates to Openreach, the internet infrastructure provider that BT's rivals would like to see spun off.
Separating the pension as part of a break-up would be a costly headache - and there is the small matter of a government guarantee on part of the scheme, something that BT fought in the courts to preserve and won.
Creating two new pension schemes, with the risk of weakening the financial resources of one or the other, might be too hot a political potato, even in the pursuit of faster broadband.
BT's share price reaction might turn out to be overdone. Ms White plans to enforce the separation by introducing not only a new board to run Openreach, but also new articles of association that will give directors a duty to serve customers of the network as much as the company's owners.
This holds out an obvious prospect of conflicts to come, and runs counter to the normal assumption that in private companies, the shareholder is king.
It also carries an echo of the "golden shares" held by the government in Rolls-Royce and BAE Systems, two companies judged important for the defence of the realm. In each case, the golden share is actually enshrined by specific articles of association that give directors and the government powers beyond those normally enjoyed by company boards.
BT's chief executive, Gavin Patterson, said he thought he could reach a compromise with Ofcom, but admitted they would need to see the detail of the new articles.
If they prove too much at odds with BT's own desire to control Openreach - which it will still own - get ready for a court battle and perhaps, eventually, a full break-up.

After EBITDA (earnings before interest, taxes, depreciation, and amortisation), the club recorded a loss of £142,659.
"It is very pleasing to report continued improvement in our results," chief executive Jamie Clifford said.
"The club has seen just short of £1m improvement in EBITDA since 2009 - an objective that was at the heart of our strategic planning."
Kent have also reported a strengthened balance sheet, with a net asset value of £7.7m - an improvement of just under £1.4m over the year.
The club also sold land at the Old Dover Road side of the St Lawrence Ground to property developers shortly after the end of the financial year, after being granted planning permission.
"This marks a key moment in the club's history. We can expect the construction of high-quality retirement apartments to commence in the coming months," Clifford added.
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.
[NEXT_CONCEPT]
The reaction from BT's investors told us much about media regulator Ofcom's ruling on the fate of Openreach, the BT subsidiary that provides much of the UK's broadband infrastructure.
[NEXT_CONCEPT]
Kent have announced an improved pre-tax profit of £345,784 for the financial year to November 2015.