Jamie's Italian in Princesshay is expected to shut in the next few months, along with five others around the country.
The vote to leave the EU is being blamed.
The chain's chief executive says it is "a tough market", which has become even harder post-Brexit.
The closures will affect 120 staff nationally, but the company says it will try to find them work in other parts of the business.
Debate over Exeter and Plymouth elections taking place
Opportunity to discuss whether elections should be postponed
Plan for 35 village homes slammed as “overdevelopment”
Traffic issues also raised
'Red' alert in Cornwall as strong winds become risk to life
Flood risk in parts of Devon too
Grit situation 'risks injuries' in Cranbrook
Many roads not 'adopted' by district council
Residents to quiz council on roundabout upgrades
A revamp of Manadon Roundabout could be on the way
North Devon foster carers could be exempt from council tax
North Devon Council looks to help its 36 foster families
