Homes and businesses will have a legal right to demand faster broadband speed by 2020, the government has said, after rejecting a voluntary offer from BT.
It has promised that the whole of the UK will have access to speeds of at least 10 Mbps by 2020.
BT, which is responsible for the infrastructure, had previously offered to carry out improvements according to its own timetable.
But the company said it accepted the government's decision.
The government believes the regulatory Universal Service Obligation offers "certainty".
Under the plan, broadband providers will face a legal requirement to provide this minimum standard to anyone requesting it, subject to a cost threshold.
The UK lags behind many countries in terms of speed and reliability.
BT customer Lee Wootton-Rowley, who lives in Wakefield, contacted the BBC to say: "I get 5 Mbps and an upload speed of 0.9 in 2017.
"I lived in Malta for four years where I was getting 60 Mbps and 20 upload speed."
Mike Simatos, from Rotherhithe in south-east London, said his broadband supply speed was no faster than 4 Mbps.
He described the service as "appalling".
Regulator Ofcom said that 4% of UK premises, or about 1.1 million, could not access broadband speeds of at least 10 Mbps.
It said poor connections were a particular concern for small businesses, with almost 230,000 unable to get a decent service.
Ofcom defines superfast broadband as a download speed of 30 Mbps or more.
Matt Hancock, minister of state for digital said on the BBC's Today programme: "Access means you can phone up somebody, ask for it and then someone has the legal duty to deliver on that promise.
"It is about having the right to demand it, so it will be an on-demand programme.
"So if you don't go on the internet, aren't interested, then you won't phone up and demand this."
In response to the announcement, BT said: "BT and Openreach want to get on with the job of making decent broadband available to everyone in the UK, so we'll continue to explore the commercial options for bringing faster speeds to those parts of the country which are hardest to reach."
Analysis: Rory Cellan-Jones, technology correspondent
In one sense, there is little new in today's announcement - people in remote places were promised a legal right to a minimum 10Mbps broadband service by 2020 and now they are going to get it.
But in rejecting BT's plan for a voluntary agreement to fill in most of the remaining parts of the country with a decent service, the government is taking quite a risk.
The plan, opposed as anti-competitive by BT's rivals, would at least have given some certainty.
But now it is far from clear who will provide this Universal Service Obligation - the government hopes new providers will come in alongside BT's Openreach - or what technology will be employed.
New fast fibre firms are now competing in the cities, but the 1.1 million homes and offices Ofcom says still cannot get a 10Mbps service are mainly in rural areas, and it is not clear they will be keen to lay cables along every lane.
Now it is the regulator's job to make sure this all works.
There are now two years to push through new legislation, work out how to police it, and determine what is a reasonable cost threshold for hooking up really remote homes.
Should be a doddle, shouldn't it?
Rival firms, which had talked of legal action if the government accepted BT's offer, welcomed the decision.
Both TalkTalk and Sky said the government had made the right decision.
Tristia Harrison, TalkTalk chief executive, said: "By opting for formal regulation rather than weaker promises, ministers are guaranteeing consumers will get the minimum speeds they need at a price they can afford," she said.
"The whole industry now needs to work together to ensure customers see the benefits as quickly as possible."
Stephen van Rooyen, Sky's UK and Ireland chief executive, said: "Government have made the right decision by choosing a fair and transparent approach that maintains competition, keeps prices fair and gives consumers a legal right to request broadband."
Following the introduction of secondary legislation early next year, it is thought it will take another two years before the right is enforced by Ofcom.
Under BT's offer, which the company had said would cost up to £600m, 98.5% of premises would have had access to a fixed broadband service in 2020.
Another 0.7% would have access to a service delivered by a combination of fixed and wireless connections.
The remaining 0.8% in the most difficult-to-reach areas would have been guided toward satellite or on-demand fibre solutions.