The pressure group has been granted permission to pursue a judicial review of the Act, on the grounds that the ban is incompatible with the right to freedom of expression under the European Convention on Human Rights.
It has gathered supporting statements from groups including Make Poverty History.
If the case succeeds, the legislation will have to be amended to allow the sort of ad currently prohibited, such as ADI's 'My mate's a primate' TV campaign.
ADI will rely on an earlier European Court of Human Rights case in which a Swiss vegetarian organisation successfully challenged a similar ban.
Because the Government is required to ensure that all new legislation is compatible with the European Convention on Human Rights, ministers have conceded the case should be heard. The hearing is expected to begin in April or May.
ADI will say the ban is too widely drafted because it covers organisations whose aim is to influence public opinion on a matter of controversy, and unfair, in that an oil company can run an advert claiming that it takes care of the environment, but environmental groups cannot respond through the same medium.
Tamsin Allen of Bindman & Partners, the law firm representing ADI, said: "We are confident the challenge will succeed."