It will house the town's library, a cafe, community space, and Anglesey's local history collection.
Built in 1855, it was home to assembly rooms and a court.
Last month, the Welsh Government successfully applied for a compulsory purchase order on the former owner of the market hall and the work is being funded by a Â£2.3m lottery grant.

Michelle Davies from law firm Eversheds echoed concerns of employers' group, the CBI that the UK government decision to cut subsidies for renewables will damage investment and jobs.
She said the Welsh government missed the boat by not doing enough when subsidies were in place.
Both government rejected the criticism.
The UK government cut subsidies because it is already meeting its renewable energy targets, but Ms Davies told BBC Radio Wales that was "not a sensible decision" because it made the UK a less attractive place to invest.
Ms Davies said Wales was not near where it should be on renewables, particularly when compared to Scotland's "strong leadership position".
"We didn't create that environment initially and we're paying the price for it now," she told the Wales at Work programme.
"Unfortunately there isn't a huge amount we can do about that now because the [UK] government sets the level of support that is provided," she said.
A Welsh government spokesman said it did not have the same energy powers as Scotland, and defended its green energy record.
The UK government said its priority was to move towards a low-carbon economy and ensure subsidies were used where they were needed most.

Summer Wroniecki broke a bone in her leg when the inflatable was blown about 150 yards into the crowd at Duffus Park in Cupar, Fife, on Saturday afternoon.
Summer had previously battled a rare type of cancer, which was diagnosed just after her third birthday.
Her father Christian wrote on Facebook that his daughter was "comfortable".
He said: "Summer is a tough cookie as you all know and will again bounce back."
Mr Wroniecki posted on Monday morning that her operation would be taking place on Tuesday.
It is believed an adult was also treated for a shoulder complaint after the incident.
Organisers said every safety precaution was risk assessed.
Holyhead's Grade II-listed market hall is to be restored after standing derelict for 10 years.
[NEXT_CONCEPT]
One of the leading lawyers in the renewable power sector has criticised the UK and Welsh governments for their approaches to green energy.
[NEXT_CONCEPT]
A six-year-old girl, who survived a cancer battle, is back in hospital after being injured when an inflatable slide blew into a crowd in high winds.