Published Monday, 05 December 2011
Mr Wilson said the act will come into effect on 14 December.
It specifically provides that asymptomatic asbestos-related pleural plaques and certain other conditions constitute actionable personal injury.
The legislation overturns a House of Lords ruling in 2007, which said victims could not claim compensation in these cases.
The corresponding law in Scotland was the subject of a long-running challenge by insurers, which came before the UK Supreme Court.
On 12 October this year, the Supreme Court rejected the insurers' claims that the legislation infringed their human rights and was outside the competence of the Scottish Parliament.
Announcing the new law on Monday, Mr Wilson said: "The challenge to the corresponding legislation in Scotland cast a long shadow.
"I fought hard to get the Northern Ireland legislation through the Assembly and submitted for Royal Assent.
"However, I always believed that the policy objectives of the Act were just and fair and that belief has now been vindicated by the ruling of the UK Supreme Court in relation to the Scottish legislation.
"The 2011 Act may be short and targeted, but it is a vitally important Act, which seeks to ensure the continued availability of a method of redress for ordinary working men and women."
Pleural plaques are small localised areas of fibrosis found within the membrane of the lung caused by asbestos exposure.
Earlier decisions had established that it was possible for damages to be awarded for negligent exposure to asbestos which had led to the presence of symptomless pleural plaques, but that was made impossible in 2007.
Mr Wilson continued: "It must be devastating to be told that you have developed a condition as a result of having been exposed to asbestos.
"To be told then that you couldn't claim compensation for that condition would clearly make the matter worse.
"That was the effect of the decision in the Johnston case and it is not surprising that people affected by the decision sought redress through Ministers and the Assembly.
"The 2011 Act essentially reverses that decision and, whilst the ability to pursue a claim in damages will not fully compensate for any wrong which was done, I hope that those affected will derive some comfort from the fact that the wrongdoer can once again be held to account."