Twenty-year-old Joe was originally refused planning permission to build at the rear of David and Lynda Carter's property in the Hankham countryside, despite there being no objections from residents or Westham Parish Council.
The Eastbourne Herald
broke the news of their disappointment, which then sparked widespread media attention with hundreds of thousands supporting Joe online on Facebook.
On Tuesday morning Wealden's head of planning, Kelvin Williams, met with the family and said an agreement had been reached.
The family will submit another planning application which will be marked for approval and heard by the council's planning committee 'as soon as possible'.
Plasterer Graham Howes, of Haywards Heath, heard Joe's story and has come forward to offer free labour.
He said, "I wanted to be able to help out and give something back, after all Joe has given up for his country."
The family has had messages from a carpet fitter and an electrician in Swindon offering free labour. The agent who has been dealing with the case, Mike Melvin, has also refused to take any money for his time.
Debi Hawkins, Joe's mum, said, "It is amazing that people are willing to give up their time.
"I am really happy for Joe but I won't be completely over the moon until we have it in black and white from the council. I will then buy Joe a spade so he can dig the first hole."
The council's rethink has been welcomed by Eastbourne MP Nigel Waterson.
"I am delighted that common sense has now prevailed and that an amended plan will go forward," said Mr Waterson.
"Joe has touched the heart of the nation. He is a young man with an incredibly positive attitude, despite the horrific nature of his injuries.
"We all owe it to Joe, and others like him, to do whatever we can to help him live as normal a life as possible.
"But I agree with the concern of the council that planning rules conflict with the clear need to provide suitable accommodation for wounded servicemen and women. I hope ministers will now take this on board."
And Mike Paine, Royal British Legion's Sussex manager, said he also welcomed the decision by the council to reconsider Joe's case.
juliet.perry@trbeckett.co.uk