The venue is trebling in size with a Â£2.7 million expansion, extending its exhibition space, café and events area, as well as opening a better shop and workspaces to let.
The building closed just over a year ago, and a date for the reopening has been fixed for late September.
"Time goes by so quickly," says artistic director Laura Sillars as she prepares to lead a tour of the extension. Sir Nicholas Serota, chair of Arts Council England, the scheme's architects and its builders are all on their way to see the work in progress which, she attests, is 'going very well'.
"People know where every Â£50 is on this project," she says, after showing off the impressively large new gallery's shiny floor, made from a granite-like, industrial material used for airport runways. "It's been so carefully budgeted and project managed."
The headline sum includes a 25-year lease on the building, so the true bill is closer to Â£1.7m. Arts Council England - which upped Site's annual funding by 60 per cent last year - gave a grant of Â£970,000, Sheffield Council pledged Â£125,000 and fundraising paid for the rest.
"We know buildings around the country that are nowhere near the same scale but have been done on 10 times the budget we've been working with," says Laura.
Site has taken over premises next door, previously occupied by Sheffield Independent Film. The new gallery is a brightly lit, stark, white cube - drawing attention to artists' work, not the room. A floor-to-ceiling window looks out on to Brown Street. When the project is finished, Site will be one of the biggest art spaces in the north, but Laura is not interested in 'a competition about size'.
"I'm interested in quality. I've brought round some major international artists to see the space and they love it. Everybody wants to show their work here, they know it is going to look amazing because the proportions, light and scale are fantastic. Big doesn't always mean better. Big can mean grand and pompous, or too much space to fill so you do the wrong things. I think this is one of the most perfect galleries in the north. We're hopefully producing something Sheffield will be proud of. We're proud of it, for sure."
An operator for the licensed café is being chosen. "We're going to get somebody who really cares about good coffee and nice food, and who'll work with the creative vibe we have here. We're looking for partners in trade, it's not just a rented block."
The café leads to Site's old gallery, now repurposed as a space for performances, events and meetings. The shop, meanwhile, will have its own dedicated area at last, offering goods from local and international makers, books, more prints and extra room in which to display them.
"There will always be something on. When the galleries are closed for the next show we'll put something on in the performance space. It's always going to be open for people."
On a practical level, new toilets are a 'big improvement'. There will be four, rather than one, and all gender-free on advice from the gallery's young people's group Society of Explorers. "It creates a sense of accessibility, welcome and inclusion."
A shared entrance has been created for the tenants behind Site, including synthpop band The Human League, and the Red Tape and Foundry recording studios. Among the gallery's tenants upstairs are digital art collective Universal Everything.
Architects DRDH, a practice with links to Sheffield, drew up the designs and Rotherham contractor-builder O&P has been carrying out the work.
"This project is not about turning us into something completely different," says Laura. "It's about embracing where we've got strengths and giving them the platform to shine."
Site specialises in contemporary multimedia and performance art. It started in Walkley as a photography gallery and moved to Brown Street in 1988. Laura joined in 2011, and co-directs alongside Judith Harry. Past exhibitors include Mike Kelley, Jeremy Deller and Sophie Calle. Site has been taking part in a cultural exchange project with South Korea - this month public art works have gone on show around the Brown Street site.
A recent visitor who had just returned from the US compared the expansion to 'a New York space in Sheffield'. "It's a really serious, proper space. There's lots of really fantastic things happening in Sheffield and this will contribute to putting it on the map for arts and culture. It's new, but it's grown out of what we already had and was maybe not as visible as it could have been. Everything we're doing, we know how to do really well, but now we have the space to do it in a way we feel more proud of."