BT will turn off its dial-up internet access service on 1 September.
The telecoms firm said it was taking the step because only a "tiny number" of its customers still went online using a dial-up modem.
It added that the vast majority of its 6.8 million broadband customers had switched to much faster connections.
However, a small number of people in rural areas where broadband will not work will struggle to get online after the change.
Dial-up customers were first informed about the impending closure in May and June this year, BT said, adding that most of these people would be able to migrate to a broadband service.
The company said that the shut-down meant about 1,000 people who lived in remote areas would not be able to move to broadband as their phone line was incapable of supporting the technology.
These people were likely to be living in some of the most remote parts of the UK, said Oliver Johnson, chief executive of broadband consultancy Point Topic.
"They will be too far from the telephone exchange to get any meaningful broadband," he said. "The distance means that the broadband signal degrades."
Those who had to stick with dial-up would still be able to get such services from BT via its Plusnet subsidiary.
Some phone lines will not support existing broadband technologies "No-one is being left without the option of an
alternative service," said a BT spokesman.
Sebastian Lahtinen from the Think Broadband news site, said the closure was a sign of the times.
"It's a statement of how mainstream broadband services have become, with entry-level broadband being cheaper than the dial-up plans BT is closing down," he said.
Dial-up or narrowband, was the technology that most Britons used to go online before home broadband became affordable. It involved modems sending data over lines more typically used for voice calls. The best dial-up modems despatched data along telephone lines at speeds of up to 56 kilobits per second, though compression could be used to improve this top speed. By contrast most broadband technologies work in the megabits-per-second range.
About 800,000 people still used dial-up in 2010, the last year for which figures were available, said an Ofcom spokesman.
"The number has now fallen so low nationally that it's quite difficult to get any accurate figures from a survey sample," he said. "We think it's in the very low hundreds of thousands but we cannot be any more confident than that."
With the vast majority of exchanges equipped to use broadband technologies such as DSL there was little reason to stick to dial-up, he said.
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