Roman, who celebrated his first birthday in July, died at home in Berwick, surrounded by his family.
His father, Stuart Greenlees, wrote on Facebook on Saturday: “It breaks my heart to say Roman passed away tonite at home.
“I am so proud of my little boy for all his fighting and proving doctors wrong all along the way. No more pain, just sleeping.”
Shortly after being born, Roman was diagnosed with Alpers disease, an incredibly rare degenerative condition.
Speaking in May, Stuart told how doctors had revised Roman’s life expectancy downwards, and how touched he and Roman’s mother, Rachel Laidlaw, had been by the town’s response.
Over £10,000 was raised for Roman, which allowed Stuart and Rachael to spend quality time with their son, and create “good, lasting memories”.
The fundraising effort for Roman was boosted by an incredible donation by the comedian Jason Manford.
The BBC star made a surprise visit to Stuart and Rachael , to offer them the profits from his Berwick shows.
Roman’s funeral is to be held at 12.30pm on Friday, at Tweedmouth Church.
Stuart wrote: “It is going to be an open service,” because he and Rachael were so struck by “the amount of support and fundraising from the community and further afield.
“If possible, men are to wear a brightly coloured tie, and women a bright scarf or handbag.”
Guests are also encouraged to bring along a brightly coloured helium balloon, to be let off together at the service.”
The couple feel this will help them to “Give Roman a good send off.”
The fundraising response meant that Stuart and Rachael were able to treat Roman to his very own sensory equipment at their Prior Park home.
The specialist equipment was installed by local tradesmen, and Roman also had his name in Disney lettering put up on one wall, with a Disney mural on another.
A multitude of events were held across Berwick to raise funds. They ranged from a sponsored leg shave to a youth disco.
Most recently a special charity gig was held at the Tavern, Eyemouth, in aid of Roman and another child, Ava Crawford of Eyemouth, who suffers from multiple sclerosis.