The first stranded Eurostar passengers are hoping to finally get moving after three days of cancellations.
Thousands were stuck after havoc was caused when snow got into the electric system of a number of trains.
More than 2,000 passengers were trapped on trains overnight in the Channel Tunnel on Friday.
Eurostar chief executive Richard Brown said that he was "very, very sorry" about the breakdowns and that modifications to the trains were being tested.
The firm was due to resume a limited service for passengers who had tickets for December 19 and 20, and then work through remaining ticket holders over the next few days.
Mr Brown said the weather in northern France, through which the 186mph trains travel, had been "unprecedented" and the worst in the company's 15-year history. This had created a new type of problem for the trains, one that the company had not experienced before.
An independent inquiry will be held into the fiasco and the team will report directly to Transport Minister Sadiq Khan.
A statement from Eurostar said: "Eurostar will be resuming a restricted service from Tuesday 22 December. Trains will depart from 07.30 to 18.00. We continue to strongly advise customers whose journeys are not essential not to travel and to change their tickets for travel on a later date or have their tickets refunded.
"Only customers who hold a ticket that was originally for travel on Saturday 19 or Sunday 20 December will be eligible for travel on Tuesday 22 December.
"Customers holding tickets for Monday 21 and Tuesday 22 December will be eligible for travel on Wednesday 23 December. Customers holding tickets for Wednesday 23 or Thursday 24 December will be eligible for travel on Thursday 24 December. Customers are advised not to attempt to travel independently via the channel ports or the Channel Tunnel unless they have confirmed bookings."
