Video report by ITV News Deputy Political Editor Chris Ship
The prime minister has held talks with European Commission president Jean-Claude Juncker and German Chancellor Angela Merkel over Brexit.
Theresa May was forced to show she is still in control of the process after the High Court ruled that parliament must have the final say on when Article 50 is triggered.
There have been calls for an emergency general election to settle the question of whether Britain withdraws from the EU.
Mrs May moved today to persuade continental leaders that the ruling will not deter her from invoking Article 50 by the end of March 2017.
A Number 10 spokesman said Mrs May explained to Mr Juncker and the German leader that "while the Government is disappointed by the judgment yesterday, we remain of the firm belief that we have strong legal arguments ahead of the case which will be moving to the Supreme Court next month".
"The Prime Minister also confirmed that the planned timetable for notification of Article 50 remains the same," the spokesman added.
Downing Street refused to say whether legislation was already being drawn up for triggering Article 50 in case the Supreme Court upheld the ruling, stressing that it was focused on winning the case.
Foreign Secretary Boris Johnson said that the will of the British people to leave the EU had been expressed "very clearly".
"It's the decision of Theresa May and of her Government to get on and make that process work," Mr Johnson said during a visit to Berlin.
The uncertainty created by the High Court judgement was underlined by Tory former chancellor and pro-European Ken Clarke, who said he would vote against invoking Article 50.
He told BBC Newsnight: "I will vote against it. I shall stick to my guns. I'm not going to cast a hypocritical vote."
Interim Ukip leader Nigel Farage suggested Mrs May call a snap general election in the wake of the ruling, since the Commons is largely pro-EU.
He told Good Morning Britain: "There is a willful attempt by a very dishonest political class to overthrow one of the major aspects of the referendum.
"My concern is if we put this back into the House of Commons, into the hands of our political class, frankly, we risk being betrayed.
"The neatest, cleanest way to finish this off is for May to call a general election on the basis that Brexit means Brexit, to get a majority and to get on with it.
"I really think you should not underestimate just how angry those 17.4 million people (who voted to leave) are going to be if Brexit doesn't really mean Brexit."
Communities Secretary Sajid Javid said on BBC Question Time that the judgement was "an attempt to frustrate the will of the British people, and it is unacceptable."
Under the Fixed Term Parliament Act, the May 2020 general election could only be brought forward if two thirds of MPs vote for it, or if the government loses a vote of no confidence and other parties do not form a government within two weeks.
After the explosive High Court ruling, Brexit secretary David Davis conceded that an act of Parliament would now be necessary to trigger Article 50.
Liberal Democrat leader Nick Clegg told ITV News that he would work with MPs from all parties to amend the government's planned legislation to ensure the public have a say on the final terms of Brexit.
Mrs May's spokeswoman said she would assure European leaders she would stick to the declared timetable for withdrawal, stating: "She will set out what the process is, which is that we are appealing and that we are carrying ahead and sticking to the timetable we have set out. This judgment is not going to derail that."
Shadow foreign secretary Emily Thornberry told the BBC: "I think that Article 50 is going to happen. I think that in the end Parliament will vote for Article 50 to be triggered.
"But the really good thing about this judgment ... is that the Government will need now to come to Parliament and actually give us some basic terms on which they are going to negotiate Brexit.
"We do not allow the executive a free hand."
Mr Juncker's spokesman made it clear to reporters in Brussels that the phone call was taking place at Mrs May's request, and not his.