The allocation was made at the start of February.
The money must be spent by the end of March 2017, according to an email seen by the BBC.
The email told medical staff to refer "long waiting patients suitable for the independent sector".
It said to refer those who require treatment for "spines, hip and knee, lower and upper limbs".
The news comes after the health minister said on Tuesday that more than £31m is required to treat patients who have been waiting more than a year for some appointments.
Michelle O'Neill said she was confident the money would be agreed after the election on 2 March.
The funding to the private sector does not include money to treat outpatients.
The email goes on to say that the £1m funding move does not affect any "in-house" funding, which the trust received in the past four months, and that staff can continue to backfill lists and carry out weekend work when teams are available.
While most people waiting may not mind when their operation is performed, it is understood that doctors and nurses would have preferred for the additional money be invested in the health service instead.
In 2015, when the then health minister allocated £40m towards waiting lists, the money could not be spent in time and only half of the amount when on operations.
That was because hospitals do not have the capacity or the staff to spend the money in such a short space of time.

The reboot of the classic Rudyard Kipling adaptation has reaped $684.8m (£468.5m) globally since its release last month.
The Huntsman: Winter's War managed takings of only $9.4m (£6.4m) in its second week of release.
Feline action comedy Keanu took third slot with a modest $9.35m (£6.39m).
Comedy Mother's Day had a weak debut, with takings of $8.3m (£5.6m), despite featuring stars including Julia Roberts, Jennifer Aniston and Kate Hudson.
All eyes are on the release of Captain America: Civil War, which opens next week in the US and has already done good business elsewhere.
Civil War opened in 37 territories over the weekend, taking in an estimated $200.2m (£136.9m).
That included record openings in Mexico, Brazil and the Philippines for the film which many believe could be this year's biggest hit.
The Belfast Health Trust has been allocated £1m in order to send patients currently on waiting lists to be treated in the private sector instead, the BBC has learned.
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Disney's The Jungle Book has topped the North American box office for the third consecutive week, taking $42.4m (£29m) according to early estimates.