The hospital, which serves more than 200,000 people in Weston-super-Mare, will close between 22:00 and 08:00 BST.
Those with serious and life-threatening emergencies are being told to dial 999 and ambulances will take them to Bristol or Taunton.
The Trust said the the temporary move was necessary until it can guarantee safe levels of staffing overnight.
It comes after a health watchdog rated its emergency care as "inadequate".
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A Care Quality Commission (CQC) report warned "significant improvements" must be made.
Medical director Dr Peter Collins described the move as "a very difficult decision" but added "it is our ability to recruit that is our challenge, not our ability to care".
But the public sector workers' union Unison warned every night the department is closed "is a risk to the people of Weston".
Christine Cook, of Unison, said the nearest alternatives were in Bristol or Taunton - some 27 miles away - would leave "sick people a long way from help and putting more strain on health staff at other hospitals".
"I'm concerned that behind the lack of answers from the trust is a government plan to further downgrade Weston Hospital.
"A similar closure in Lincolnshire has been extended to over a year. If that happens here, related departments like the intensive care unit will be affected too."
Weston General Hospital's A&E department closes overnight later because it cannot be staffed safely.