It was due to start in the Court of Appeal this morning before the Lord Chief Justice Sir Declan Morgan.
But following the intervention of Attorney General John Larkin QC, the case has now been pencilled in for Monday, May 9.
The company was last year ordered to pay £500 for refusing to make a cake with a pro-gay marriage slogan on it.
A Belfast County court judge found that customer Gareth Lee had been discriminated against on the grounds of his sexual orientation.
He was backed in the case by the NI Equality Commission.
The commission paid almost £39,000 in legal fees.
The judge had ruled that, as a business, Ashers was not exempt from discrimination law.
The firm was found to have discriminated against Mr Lee on the grounds of sexual orientation as well as his political beliefs.
At the time, the judge said she accepted that Ashers had “genuine and deeply held” religious views, but said the business was not above the law.
However, the McArthur family who run Ashers, said they had decided “after much careful and prayerful consideration” to appeal.
In a statement before going into the High Court this morning, the McArthur family said:
We are looking forward to this opportunity to have our case heard by the Lord Chief Justice because we believe the county court got the original ruling wrong.
As I have said many times before, Ashers Baking Company does not discriminate against anyone.
We took issue with the message on the cake and not the customer and as a family we do believe we should retain the freedom to decline business that would force us to promote a cause with which we profoundly disagree.
As Christians we cannot simply switch off our faith when we enter the workplace on a Monday morning.
To be a Christian at all is to strive to live for Christ in every corner of our lives.
We served Mr Lee as we would any other customer.
We were simply unwilling to endorse a campaign for a new law that so clearly goes against what the Bible says about marriage.
And for that we were punished.
Christians are law-abiding citizens and we expect the law to protect us as much as anyone else.
We hope that the judicial system will now make the correct decision and protect our freedom to carry out our work without being forced to violate our consciences.
As a family we have found the whole legal process very difficult.
We would rather not have to be here today. But we knew that we had to appeal, not only on our own behalf, but on behalf of other family businesses who could be forced to endorse or promote views with which they profoundly disagree.
So today we appeal to the Lord Chief Justice and his colleagues to overturn the county court ruling.
We appeal to them to recognise that there is a big difference between refusing to serve someone because of their sexual orientation or political opinions, and choosing not to endorse those ideas.
We appeal to those who would condemn us for our actions to consider what they would have done if they were required by law to use their creative abilities to help promote a cause which went against their strongest convictions.
And we appeal to our fellow Christians, to have courage in the face of such challenges to our faith. We thank them for their prayers and ask them to continue to pray during the hearing and as the judges consider their ruling.
We know that most people do not think we have done anything wrong and we are grateful for the support of the many people who, though they disagree with us on marriage, recognise that we should not be forced to deny our beliefs.
Above all we continue to put our trust in God.
He is faithful. He is in control. His will be done.