Mrs May said the Government would work with the public to "raise awareness" of the dangers of driving and dialling.
She said punishments for drivers who kill and maim people because their attention is on their phone should be made to "fit the crime", as a deterrent to other motorists.
Tougher penalties are due to take effect next year, thanks in part to pressure from the RAC.
RAC research released in September showed that the number of motorists that think it is acceptable to make a quick call doubled from 7% in 2014, to 14% in 2016.
The organisation’s road safety spokesman Pete Williams said: “Better enforcement needs to be backed up by more driver education about the true dangers of handheld mobile phone use, and a heavyweight road safety campaign akin to that which has been successful in making drink-driving socially unacceptable.”
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Speaking during a visit to India, Mrs May said: "Sadly we have seen too many times the devastating and heart-breaking consequences of using a mobile phone while driving. A moment's distraction can wreck the lives of others forever.”
The Prime Minister’s comments come after the DfT's response to a consultation on the punishments handed out to motorists caught using a hand-held phone.
The document confirmed the Government's intention to double the fine from £100 to £200 and increase the penalty points from three to six, which first emerged in September.
First time offenders will also no longer have the option of taking an educational course to avoid getting points on their licence.
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It comes shortly after the jailing of lorry driver Tomasz Kroker, who was scrolling through music on his phone just before he ploughed into stationary traffic on the A34 in Berkshire, killing Tracy Houghton, 45, her sons Ethan, 13, and 11-year-old Josh, and step-daughter Aimee Goldsmith, 11.
DfT figures show that a driver impaired or distracted by their phone was a contributory factor in 440 accidents in Britain last year, including 22 which were fatal and 75 classed as serious.
The RAC praised the Prime Minister’s comments saying tougher penalties must be paired with better driver education.
Mr Williams said: “We welcome stiffer penalties for handheld mobile phone use and believe this will send a very strong message to motorists. However, we believe this has to be done in conjunction with a heavyweight road safety campaign so we therefore welcome initial proposals for a new THINK! initiative.”
The RAC’s Report on Motoring 2016 found that the problem is at “epidemic proportions” as almost half (48%) of motorists admitted to using a handheld phone at the wheel to talk, text or use other apps in the last year.