I have a question that is very troubling. One of my friends was a teacher who taught disabled children for two years. She related a case to me of a mentally handicapped girl who was impregnated by her brother. The baby was born so disabled that it had to be institutionalized. My friend said that the babies from such pregnancies are normally born that way. I’m sure there must be many others out there who are faced with this dilemma.
 
She seemed to feel it would be better that the girl be surgically treated to prevent such a pregnancy. I find this very troubling because I am completely pro-life. I’m sure there are many other cases where girls are taken advantage of and mothers are faced with this dilemma.
 
I would appreciate if you could discover the Church's stand on this.

What you describe is both very real and very sad.  Lord, have mercy on us all!  What is called for in this situation is vigilant oversight on the part of parents. In my experience as a priest and as a confessor, it is never advisable for a minor to be alone out of public view with an older person, even if they are related. I know, I know. That’s quite a statement, but the wound of original sin runs deep in human nature.
 
As to your question, even in the case of children with serious developmental disabilities such as Down syndrome, it is never permitted to sterilize a person by vasectomy or tubal ligation. Such acts of contraceptive sterilization are intrinsically evil.  No matter how praiseworthy the intention, you can never justify an intrinsically evil act. 
 
So, what is to be done if a Down syndrome girl becomes pregnant?  First, give her very good care and support.  Second, provide for the safe delivery and support of the child.  Third, assign and enforce – to the degree possible – financial obligations on the father of the child at the same time as criminal penalties are determined by a court of law.
 
And pray.