BBC Wales has learned an agreement has been struck to end the dispute.
It had been claimed a spelling error by officials at Companies House in Cardiff had led to the demise of engineering firm Taylor & Sons.
Companies House records had been amended to show Taylor & Sons had been wound up in 2009.
But the failing business was another firm called Taylor & Son - with "Son" in the singular.
Taylor & Sons dated back to 1875 and more than 250 people lost their jobs when it went under, and former co-owner, Philip Davison-Sebry, claimed Â£8.8m in damages over the mix-up.
In January 2015 the High Court judge ruled in Mr Davison-Sebry's favour and found Companies House, when entering a winding up order, owed reasonable care to ensure the order was not registered against the wrong company.
Companies House began an appeal against the ruling in January last year, arguing the case set a "dangerous precedent".
A full appeal was listed for this month, but BBC Wales has learned an agreement was reached last spring.
A spokesperson for Companies House said: "We have agreed confidential terms of settlement and this case has now concluded."
No further details are being disclosed by either party.

The man, who was 13 when he attended St William's in Market Weighton, told the jury James Carragher used to make him "upset" before he was abused.
The ex-principal denies 50 counts of indecent assault between the 1970s at the East Riding of Yorkshire site.
At Leeds Crown Court, Mr Carragher also denies 12 other serious sex offences.
Earlier in the trial, the jury heard the 75-year-old, of Merseyside, was jailed for seven years in 1993 and 14 years in 2004 for sex offences.
The home for boys with behavioural problems was run by the De La Salle order. It has since closed.
Giving evidence behind a screen for legal reasons, the former pupil said he was sexually abused by Mr Carragher on "a number of occasions".
"He would get me upset to get me to start crying. He put his arm around me trying to comfort me and would then start touching me," he said.
Richard Wright QC, prosecuting, asked: "Why did you not go to the police sooner?"
He replied: "I was ashamed. Even though I was only a child, I still let him do it."
The man said he had not discussed the matter for 45 years.
Mr Carragher is on trial along with former chaplain at St William's, Anthony McCallen, 69, and former teacher Michael Curran, 62.
The jury was previously told how Mr McCallen was convicted of abusing two boys in the 1990s.
Mr Curran has no previous convictions.
Mr McCallen, of Merseyside, denies 18 indecent assaults and seven other serious sexual offences.
Mr Curran, of Teesside, denies one count of assault occasioning actual bodily harm and another of indecent assault.
The trial continues.

The bill against the practice, which aims to "cure" a non-heterosexual person of their sexuality, was voted through unanimously.
Under the new law, anyone who tries to "change, repress or eliminate a person's sexual orientation, gender identity and/or gender expression" will be fined or even jailed.
Professionals will face heftier fines of up to 10,000 euros (Â£8,450/$10,700).
They could also be jailed for up to a year, according to Malta Today.
The bill also enshrines in law that "no sexual orientation, gender identity or gender expression constitutes a disorder, disease or shortcoming of any sort".
Malta was named the best European country for LGBT rights by advocacy group ILGA-Europe in 2015.
Gay conversion therapy has increasingly come under the spotlight in recent years, but remains more popular in the US than in Europe.
Its supporters claim it uses standard psycho-therapeutic and counselling techniques so people can change or reduce their "homosexual tendencies" of their own free will.
But the World Psychiatric Association has denounced the practice as unethical, unscientific and harmful to those who undergo it.
Two years ago, NHS England and the Royal College of Psychiatrists - along with 12 other organisations in the UK - signed an agreement which called it "potentially harmful and unethical".
It has also been banned on minors in several places in the US, including California and Illinois.
A multi-million pound court battle over a spelling mistake said to have led to the collapse of a Cardiff company has been settled.
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A former pupil at a Catholic home and school for delinquent boys told a court he could not report his alleged abuser to police because he felt "ashamed".
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Malta has become the first European country to ban gay conversion therapy.