Swansea Bay University Health Board (SBUHB) said the delay would last until the new financial year in April.
It comes after waiting times for A&E departments in Wales hit their worst levels in December.
The health board said the decision does not affect the recruitment of doctors or nurses or “any operationally critical posts”.
People who have already been offered a start date will not be affected, it added.
Earlier this month, Health Minister Vaughan Gething said winter pressures were “exceptional”.
Cardiff and Vale Health Board has also put a freeze on the recruitment of “non-clinical” staff until April, but did not put this down to financial constraints.
It said it had “worked hard to reduce its deficit and is projected to break even for the 2019-20 financial year”.
Angela Burns, the Welsh Conservatives’ shadow health minister, said she was “really concerned”.
She said: “Too many health boards have fallen back on relying on agency staff across all roles resulting in a temporary, high-cost solution which only papers over the cracks.”
SBUHB has a £12.3m deficit for 2019-20, £2.4m more than the previous year.
As a result, the recruitment of non-rostered staff – mainly in clerical, administrative and back office roles – has been put on hold.
It said it was “prioritising our spend on delivering high-quality and safe healthcare”.
Ceri Phillips, professor of health economics at Swansea University, questioned how the recruitment freeze would hit NHS services.
“Everyone employed by the NHS should have a role to play in patient care, if not, why are we employing them in the first place?” he asked.
“If some of these clerical staff are important to manage patient flows through the system then you are going to see an exacerbation of figures we have seen in A&E departments, for example.
“The situation facing health boards is that 70-75% of their expenditure goes on wages and salaries.
“Coupled with the cost of agency fees across Wales, that represents a massive amount of money.
“If they got the workforce right and the number of people right at the time then we wouldn’t see the same pressures to the same extent.”
SBUHB also confirmed some clerical and administrative agency staff were “temporarily stood down” over the two-week Christmas period – the only health board in Wales to do this.
It said agency staff were given the option to take annual leave instead of being stood down, if they had it available.
But sources within the health board claimed agency workers were only told in early December they would be stood down, leaving them with no time to accrue paid leave.
It is understood several members of staff had concerns this would leave them financially short over Christmas.
Hazel Robinson, director of workforce and operation development at SBUHB, said “urgent action to reduce costs” needed to be taken “while still safeguarding patient services”.
She said the aim to break even would not be possible with “continually high operational pressures”.
Only in “exceptional circumstances” will new staff be allowed to take up posts before the new financial year, she added.
Of the decision to stand down staff over Christmas, she said: “This was not a decision taken lightly, but weighed against the need to safeguard money for patient services.”
Hywel Dda health board and Powys Teaching Health Board said they were both “actively” recruiting and, in some cases, struggling to fill current vacancies.