Eurostar has apologised to passengers and announced it is hoping to run services again on Tuesday after a three-day shutdown.
The high-speed rail company's chief executive Richard Brown said he was "very, very sorry" about what had happened.
He also warned that people travelling in the next few days would, for the time, face "a limited service".
He said the weather in northern France that has brought travel chaos to thousands of passengers was "unprecedented" in the company's 15-year history.
Mr Brown admitted that Eurostar would have "a lot of work to do" to reassure customers that the company was worth travelling with.
Speaking at Eurostar's London terminal at St Pancras station, Mr Brown said that test trains had run today to see whether modifications made by engineers were working. The modifications involve preventing snow getting into the electrics of the 186mph trains that run between London and Paris and Brussels.
He added that the first priority when services resumed would be people who had been due to get away last weekend.
His comments came after Eurostar announced it had commissioned an independent review into the problems which left thousands of passengers stranded.
Earlier, transport minister Sadiq Khan said he was "angry" that passengers had still not been told what was going on, and that he and his French counterpart had demanded to know what had gone wrong.
Channel Tunnel train shuttle company Eurotunnel had to suspend access to its Folkestone terminal at Folkestone in Kent after being inundated with passengers. Even those with day-trip bookings were turned away.
