The council went back to the market last September after failing to agree a final price with original preferred contractor Kier.
The council said: “As a result of this process the total required budget for the project has now reduced from £41.6m to £32m.”
Council leader Cllr Steve Count said: “I’m pleased the bold decision to re-tender the design and build contract was taken and it has clearly paid off, almost £10 million has been saved, compared to if we had awarded the previous contract in August last year.
“Obviously, we will now need to work with Jones Brothers and consider the impact Covid-19 might have on the situation but we are still aiming to start on site by the end of the year, as promised.”
Jones Bros contracts director Geraint Thomas said: “We’re thrilled to have been appointed to deliver this scheme.
“We are aware how important this is to the area and we are looking forward to starting work on site in due course.
“As with all Jones Bros projects, we will be aiming to boost the local economy during construction, with opportunities for local suppliers as well as recruitment of apprentices and experienced operators.”
A specific Covid-19 project contingency budget of £1.5m to fund any additional costs directly associated with the project caused by the impact of coronavirus has been agreed.
The next steps will now be to finalise the design and make preparations to start construction later in the year.