Justice secretary Liz Truss said the independence of the judiciary was the "foundation upon which our rule of law is built" after attacks on the legal system following the High Court's ruling on Article 50.
However she did not condemn attacks made on the profession from some Conservative MPs and media outlets in the wake of Thursday's ruling.
"Our judiciary is rightly respected the world over for its independence and impartiality," she said.
"In relation to the case heard in the High Court, the Government has made it clear it will appeal to the Supreme Court. Legal process must be followed."
Liz Truss has been appointed justice secretary in Theresa May's new cabinet.
Ms Truss takes up the position formerly held by Michael Gove, who has been sacked from the Cabinet.
Arriving at the Ministry of Justice,Truss said she was "delighted" at being appointed to the role, and asked if she had any plans, she said: "Well, it's early days."
Teachers have been given the "green light" to "get tougher on discipline", an education minister told Daybreak.
Liz Truss said one third of teachers were "not sure" how to discipline children and the Government was supplying teachers with clear guidelines of what punishments could be administered.
Education minister Liz Truss has called for a shift in attitude towards seeing childcare as a profession that requires training.
She said: "It is a very demanding job that requires great and specialist expertise. I am not trained to do the job. I'm a politician, not an early educator. And I'm equally sure I could not walk into a class of 30 14-year-olds and teach them German."
Education minister Liz Truss has defended her plans to raise the qualifications and pay of carers for young children in England. She told Daybreak that childcare should not be seen as "babysitting".
The education minister Liz Truss has responded to questions on Twitter saying that England pays carers some of the lowest wages in Europe, and has among the lowest qualification thresholds.
She also responded to suggestions that very young children need "love" rather than education.
Nurseries and childminders will be allowed to look after more children as part of coalition efforts to cut childcare costs.
Education minister Liz Truss will announce today that the ratio of adult to children will be relaxed where the carers' qualifications meet new standards.
Those staff deemed qualified enough will be able to take charge of six two-year-olds rather than four while the ratio for under-ones will go up from three to four.
The changes, which have been criticised as lowering standards, will be brought in alongside reforms that will see higher qualifications required of those caring for pre-schoolers.