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.
Devon man to run round mainland Britain
Steve Jones will be first man over 60 to do so, if he makes it
Decision on Exeter swimming pool due this week
Protests gather pace
South Hams tourism down, but average spending up
Sector is worth £266 million to the district
More East Devon council workers leaving
One in eight quit last year
Dartington pool expected to reopen
Developers stumping up £88,000
Honiton man jailed for terrorism-related crimes
Tobias Gleed "is a threat" say counter-terror police
