Strathclyde Partnership for Transport (SPT) said all services would be suspended from 2 July. They are due to resume on 1 August.
A replacement bus service will be provided for subway passengers during the closure.
It will follow the subway route, with additional direct services to the city centre from key stations.
They include park and ride facilities at Bridge Street, Kelvinbridge and Shields Road. Services will run every five minutes at peak times and every 10 minutes during off-peak.
All passengers will pay £1 for a single journey, £2 for a return and £2.50 for an all-day ticket. Child fares will be 50p for a single and £1 for a return.
The modernisation work involves replacing a 40-year-old section known as "ramps and turnouts", which permits trains to access the tunnels from the surface depot at Broomloan and cross over between the inner and outer circles.
SPT said the section had to be fully replaced to ensure a reliable subway service in the future.
SPT chairman Jonathan Findlay said: "To date, SPT has achieved all of the modernisation works without any disruption to passenger service which is a considerable achievement.
"We have now reached a crucial stage in the modernisation plan that requires us - for reasons of safety for everyone involved - to suspend services for a short time during July.
"We apologise for any inconvenience to all our passengers but our full replacement bus service will mean that they can still get to where they want to go."
He added: "All our passengers are encouraged to turn up at their designated stop, allow a little longer for the journey, and leave the rest to us."
All the information on the subway suspension and the replacement bus service is available on the SPT website.
The Glasgow subway is the third oldest underground system in the world and is 120 years old this year.
It is currently undergoing a £288m modernisation plan to upgrade or replace trains, signalling, platforms and stations.
The Scottish government is contributing £246m towards the cost.

The 18-year-old centre-back came through the youth system at Stoke City and impressed during a trial with the Latics.
The contract also includes the option of an extra year that Oldham can take up after this season.
"He is really athletic, only 18 and definitely one for the future," boss David Dunn told the club website.
"He has a lot of very good attributes and we spoke to Mark Bowen at Stoke City about him so we have decided he is a good young signing for us."
Find all the latest football transfers on our dedicated page.

According to its website, both the 32GB and 64GB models are "currently unavailable", and Amazon says it has no plans to replenish its inventory.
It follows reports of lay-offs at the US company, after the phone failed to resonate with consumers.
Amazon has not responded to requests from the BBC for comment about its plans for the phone.
Launched with much fanfare in June 2014, the Fire Phone was Amazon's first foray into the smartphone market.
The main selling point was its "dynamic perspective" camera, which can track the user's movements and give the impression of depth and 3D.
It also includes a 24-hour customer service tool and an in-built encyclopaedia called X-Ray.
But the rollout quickly faltered, as reviewers took against the phone, and Amazon cut its price drastically last September.
In October 2014, the company revealed it had suffered a $170m (Â£110.5m) writedown "primarily related" to the gadget.
And last month, the Wall Street Journal reported it had laid off "dozens" of Fire Phone engineers at its Silicon Valley research-and-development lab.
Stuart Miles, editor and founder of technology news website Pocket-lint, said he was not surprised by the decision to pull the phone.
"The phone industry is incredibly competitive, and it's very foolhardy for any company to believe you can come in on your first product and make a difference.
"Amazon obviously believed it could launch something, which unfortunately didn't gain any excitement.
"The dynamic perspective camera was pretty much seen as a gimmick which didn't really add anything to the overall experience."
Amazon is still selling its tablets, such as the Kindle and Fire. On Tuesday, the Wall Street Journal reported Amazon planned to launch a cut-price $50 (Â£33) tablet this Christmas, along with new tablets with 8in (20cm) and 10in screens.
The Glasgow Subway is to close for about four weeks in the summer while modernisation work is carried out.
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League One side Oldham Athletic have signed defender Theo Vassell on a deal until the end of the season.
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Online retailer Amazon has stopped selling its Fire smartphone, just 15 months after the handset was launched.