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."

A man was arrested after the fires at Wasserstein and police suspect arson. No-one in the hostels was badly hurt.
More than a million refugees and migrants have arrived in Germany this year, and for most their first stop has been in Bavaria.
President Joachim Gauck devoted his Christmas message to the recent influx.
"We have shown what we are capable of, in terms of good will and professionalism, but also in the art of improvisation," he said, praising fellow Germans for their response to the crisis.
But he also called for the full force of the law to be used against anyone using violence against the new arrivals.
"Arson and attacks on defenceless people deserve our contempt and deserve punishment," the president said.
Refugee hostels in Germany have come under more than 200 attacks this year alone, according to research by Die Zeit newspaper, but the authorities have secured convictions against only a handful of attackers.
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Chancellor Angela Merkel promised last week that Germany would "noticeably reduce the number of refugees" in response to criticism from within her centre-right CDU party.
Up to 40% of arrivals in Germany have come from Balkan countries such as Albania and Kosovo which are considered safe. Most of the other arrivals are refugees from Syria, Iraq and Afghanistan.
European countries have struggled to cope with the large number of people reaching the shores of the EU. This week, international migration officials said more than a million "irregular" migrants and refugees had come by land and sea, the vast majority via Greece.
Czech Prime Minister Bohuslav Sobotka singled out Mrs Merkel for blame for sending out a signal across the Middle East and Africa which "stimulated illegal migration to Europe".
UN refugee officials say 3,716 people died or went missing this year as they tried to cross into Europe.
But there was good news for one Afghan family who feared their 10-year-son had drowned off the Turkish coast.
The Rabani family, who have since arrived in the German state of Lower Saxony, became separated from their son as they boarded a boat in Turkey. His boat capsized and many of those on board drowned.
A German Red Cross official eventually succeeded in tracking down the child, who was found staying at a refugee hostel in Switzerland. The family hope to be reunited in a few weeks' time.
In a separate development, a young Palestinian refugee who made headlines in July when she cried in front of Mrs Merkel has reportedly been told by German immigration officials that she and her family will be able to stay in the country until 2017.
Reem Sahwil, 14, had told the chancellor on live TV that she wanted to go to university but she burst into tears when the chancellor explained that not all migrants could stay in Germany and that politics "can be tough".
Mrs Merkel tried to comfort the girl by stroking her head.
Weston General Hospital's A&E department closes overnight later because it cannot be staffed safely.
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Twelve people were hurt, including seven teenagers, when fires broke out at two neighbouring refugee hostels in the south German region of Bavaria.