The ruling as to whether the Prime Minister has the right to trigger Article 50 without Parliamentary approval is to be delivered on Thursday.
The historic case has been brought to bear by Gina Miller, an investment fund manager and philanthropist living in London.
Miller contends that Theresa May has no legal power to start negotiations with the European Union without a Parliamentary act - something she intends doing by the end of March 2017.
Over the course of the case, judges heard submissions that the Government does not have the power to exercise royal prerogative to begin the process of removing statutory rights granted to UK citizens under the European Communities Act 1972 - an act which integrated the EU into the unwritten UK constitution.
On the other hand, lawyers acting on behalf of the Government say that if the challenge succeeds, they ''could not give effect to the will and decision of the people, as clearly expressed in the referendum, to withdraw from the EU without further primary legislation''.
In final submissions, they revealed that any new treaty agreement with the EU following Brexit would "very likely" have to be ratified by Parliament.
The announcement will be made at 10am in London by Lord Chief Justice Lord Thomas, Master of the Rolls Sir Terence Etherton and Lord Justice Sales.