The company has already been given two extensions since the franchising programme collapsed two years ago.
This extension enables a major rail electrification programme to be completed but it has angered the Rail Maritime and Transport (RMT) union.
The franchise includes services from London Paddington to South Wales.
It also passes through the Cotswolds and the West of England.
Brendon Kelly, from the RMT, said: "All the investment at current is coming from the government, nothing is coming from the private operators, they're literally just managing it and sharing out the dividends to their shareholders.
"That's money we could be spending on the railways."
Some £5bn is being invested to electrify the track between Bristol and Cardiff, which involves major work such as raising road bridges.
The BBC understands the plan is to complete the electrification project, get the trains running and then put out a new franchise for the railway open to all bidders.
Chris Irwin, from passenger lobby group Travel Watch South West, said: "One of the worries was that before the government decided to look at the possibility of the five-year franchise extension, we'd get a new franchise entrant coming in.
"That new franchise entrant would lack the inside knowledge of how this particular bit of railway works and that would be a real disaster."
A Department for Transport spokesman said: "We are committed to providing better services, more seats and more trains across the rail network in the South West as part of our long-term economic plan.
"As part of that process we are currently negotiating a directly awarded contract for First Great Western to continue running services. We have not yet decided how long that contract will be."