The British currency has been falling heavily in the past few days. May's last-ditch attempt to get backing for a Brexit deal triggered a revolt on Wednesday by some of her Brexit-supporting ministers, deepening the political uncertainty in Britain.
House of Commons leader Andrea Leadsom resigned and the BBC said more ministers could follow.
"What comes next is wholly uncertain, but most likely market pricing of a No Deal Brexit will increase this summer. Despite thirteen consecutive days of decline against the EUR, GBP still remains fragile. Cable remains on course for $1.25," ING analysts said in a note.
The pound dropped 0.4pc to a new 4-1/2 month low of $1.2605 by 0740 GMT. It is the worst performing major currency in May, and so far has lost 3.3pc of its value against the dollar.
Against the euro, the British currency weakened 0.3pc to 88.385 pence, just below Wednesday's low - should the pound end Thursday down again, that would mean 14 consecutive days of losses against the single currency, its longest losing streak on record.
Nearly three years after the United Kingdom voted 52pc to 48pc to leave the EU, it remains unclear how, when or even if it will leave the club it joined in 1973.
The current deadline to leave is Oct. 31.
When May departs, her Conservative Party will elect a leader who is likely to want to renegotiate the deal May agreed with the EU in November, raising the chances of a confrontation with the bloc.
The bookmakers' favourite to succeed May is Boris Johnson, the face of the official campaign to leave the EU, who has said he wants a more decisive split from the bloc.
British voters head to the ballot box on Thursday for European parliamentary elections, and polling data published before polls opened put Nigel Farage's Brexit Party on course to win and May's Conservatives to do very badly.