Daniel McArthur (26), the firm's general manager, said they had been buoyed by the level of support.
He said: "It has been a long and difficult road, but we have been sustained every step of the way by the word of God and by the many thousands of people who have supported us.
"Many share our beliefs about marriage. Many more defend our freedom to hold those beliefs.
"As a family, we are simply hoping and praying for a just outcome so that our ordeal in court next week will be our last."
Northern Ireland's most senior judge Lord Chief Justice Sir Declan Morgan and two other top judges are due to hear the case at Belfast High Court on Monday - exactly two years after the controversial order was placed at Ashers' city centre branch on May 9, 2014. It is expected to last several days.
The Northern Ireland Equality Commission, which monitors compliance with the region's anti-discrimination laws, took the landmark civil action on behalf of Gareth Lee, a gay rights activist and member of the LGBT advocacy group Queer Space.
Mr Lee had wanted a cake featuring Sesame Street puppets Bert and Ernie with the slogan 'Support Gay Marriage' for a private function marking International Day Against Homophobia in May 2014.
He paid the £36.50 price in full but was phoned two days later and told the company could not fulfil the order.
The high-profile case was heard at Belfast's County Court over three days last March.
Giving evidence, Ashers' owner Karen McArthur said as a born-again Christian she knew in her heart she could not make the cake, but had taken the order to avoid a confrontation in the shop.
Daniel McArthur told the court his family could not compromise their religious beliefs, despite the legal ramifications, while Mr Lee claimed he was left feeling like a lesser person.
Delivering her findings, District Judge Isobel Brownlie said the bakers had breached equality legislation and had directly discriminated against Mr Lee. Ordering Ashers to pay agreed damages of £500, the judge said religious beliefs could not dictate the law.
An appeal had been scheduled for February but proceedings were dramatically halted after a last minute intervention from Northern Ireland's Attorney General John Larkin QC, who advises Stormont politicians on legal matters.
Throughout their legal battle Ashers have been supported by The Christian Institute which has organised public rallies and garnered financial backing for the case.
Simon Calvert from the Institute said: "We have a strong case and are looking forward to being able to put that case in court next week.
"We know there are huge numbers of people around the country who, even if they don't support the McArthurs' beliefs on marriage, do not think they should be compelled to promote views that go against those beliefs. We're hopeful that the Lord Chief Justice and his colleagues on the bench will take the same view."