The incident happened when Clare Walker, 29, was feeding 15-month-old Brayden through a tube.
The little boy suffers from a serious kidney condition called posterior urethral valves and so has to be fed milk in very precise quantities via a special machine.
But when Clare tried to feed him in the Bon Accord Centre Costa branch in Aberdeen she says she was told by staff to leave – and that the manager warned her not to return.
Clare said: "Brayden gets air through it when he feeds. "A woman near us told staff he was putting her and her friend off their coffee as they could hear the wind in his tube.
Nobody bats an eyelid when a woman breastfeeds in public – but when you feed a child through a tube they're outraged.
"I was sickened that someone could be so shallow."
Claire, who is married to Kenneth, 44, said she had not yet been contacted by Costa and no apology had been given.
The Equality and Human Rights Commission for Scotland said that they would look into any discrimination claims brought forward by the family.
A spokeswoman said: "In cases like this we would encourage Claire and Kenneth to get in touch with our helpline and we can assess the situation from there."
Costa said they were looking into the incident.
A spokeswoman said: "We have been trying to contact the lady to get the full details because we can't take any other action until we know which of the four Aberdeen shops she was in.
"We are looking into this isolated incident as a matter of priority, but in the meantime we apologise for any distress this obviously caused the lady in question."
Costa Coffee later gave this statement:
"We are able to confirm that we have now spoken to Mrs. Walker, the lady involved, and sincerely apologised for the incident involving her family.
During our conversation we agreed the following three steps would take place immediately:
We will deal with the staff member involved in line with our Human Resource procedures.
We will reinforce our policies around disability to all Costa team members at the earliest opportunity.
We will increase our focus on teaching staff how to deal better with sensitive situations whatever they may be.
Mrs. Walker has provided the following comment. "I felt the whole situation was disrespectful but I am pleased that Costa called me to apologise. I agree that this is a one-off, isolated situation, caused by one person and not Costa as an organisation."'
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