Following a successful trial launched in the summer, the chain has expanded the service to 96 stores across the UK, including the branches at Sandpiper Drive in Leith and Straiton Retail Park.
The service gives customers access to Asda’s full online range of more than 30,000 grocery products for delivery within the hour if they live within a three-mile radius of a store in the scheme and order up to 70 items.
But the cost for each delivery will be £8.50 regardless of how many items are ordered.
Asda also recently extended its partnership with Uber Eats to more than 300 stores, although the number of products available is far smaller.
If the latest stage of the rollout is successful, bosses hope to expand it further.
Simon Gregg, Asda’s vice president of online grocery, said: “We are rolling out our express delivery service to almost 100 stores after a trial showed there was a clear gap in the market for a speedy delivery service offering our full online product range for delivery within one hour.
“As the only grocer to offer this service, we have seen a significant number of customers access this option, with slots regularly selling out at our pilot stores.”
He also pointed out that Asda recently announced a partnership with the tech firm Wayve to test driverless vans.
Rivals including Morrisons, Tesco and Sainsbury’s already offer one-hour – or faster – delivery services through Deliveroo, Uber Eats, Amazon and other providers – although none offer a full range of products.
Tesco, which had previously not offered a rapid delivery service, launched its own trial, called Whoosh, in Wolverhampton and has plans to expand it.
Sainsbury’s also operates its own service, called Chop Chop, which it expanded to 50 stores in 20 cities last year.
Waitrose announced plans to shut down its own rapid delivery service, saying it would instead focus on expanding its partnership with Deliveroo.
Morrisons offers same-day rapid deliveries as part of its tie-up with Amazon, which also offers customers its own Amazon Fresh products.