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The White House initially said that the 14-year-old had tested negative, after both of his parents tested positive earlier this month.
However, Melania said on Wednesday that subsequent testing showed Barron had also come down with Covid-19.
She revealed her son's condition in an article she wrote that was published today.
It was a fortnight ago that she and President Donald Trump tested positive for Covid-19, she said.
"Naturally my mind went immediately to our son. To our great relief he tested negative, but again, as so many parents have thought over the past several months, I couldn't help but think 'what about tomorrow or the next day?'. My fear came true when he was tested again and it came up positive.
"Luckily he is a strong teenager and exhibited no symptoms. In one way I was glad the three of us went through this at the same time so we could take care of one another and spend time together. He has since tested negative."
Melania Trump said she was fortunate that her infection came with "minimal symptoms".
"I experienced body aches, a cough and headaches, and felt extremely tired most of the time. I chose to go a more natural route in terms of medicine, opting more for vitamins and healthy food.
"We had wonderful caretakers around us and we will be forever grateful for the medical care and professional discretion we received from [White House doctor Sean] Conley and his team.
"It was an unfamiliar feeling for me to be the patient instead of a person trying to encourage our nation to stay healthy and safe. It was me being taken care of now, and getting first-hand experience with all that Covid-19 can do.
"As the patient, and the person benefiting from so much medical support, I found myself even more grateful and in awe of caretakers and first responders everywhere.
"To the medical staff and the residence staff who have been taking care of our family—thank you doesn't say enough."
The First Lady said she had now tested negative for Covid.
"I am happy to report that I have tested negative and hope to resume my duties as soon as I can. Along with this good news, I want people to know that I understand just how fortunate my family is to have received the kind of care that we did.
"If you are sick, or if you have a loved one who is sick - I am thinking of you and will be thinking of you every day. I pray for our country and I pray for everyone who is grappling with Covid-19 and any other illnesses or challenges.
"I encourage everyone to continue to live the healthiest life they can. A balanced diet, fresh air, and vitamins really are vital to keep our bodies healthy. For your complete well-being, compassion and humility are just as important in keeping our minds strong.
"For me personally, the most impactful part of my recovery was the opportunity to reflect on many things - family, friendships, my work, and staying true to who you are.
"I want people to know that I understand just how fortunate my family is to have received the kind of care that we did. If you are sick, or if you have a loved one who is sick - I am thinking of you and will be thinking of you every day."