Dealing with a diagnosis (or possible diagnosis) of Down's Syndrome can be devastating. It is important that you ask your child's pediatrician
all of your questions and concerns.
As you suggest, diagnosing Down's Syndrome is done on a genetic basis, since the syndrome itself is caused by an extra chromosome. All of our cells are supposed to have 46 chromosomes that are found in pairs. There are 23 different human chromosomes, and each cell is supposed to have two of each kind. Down's Syndrome is one of a handful of known genetic disorders in which having an extra chromosome causes a syndrome. in the case of Down's, affected patients have an extra chromosome 21--this is why the syndrome is sometimes called trisomy 21. Diagnosing Down's requires identifying that extra chromosome in cells taken either from a baby or from the placenta.
If your baby is found to have Down's syndrome, he or she may need some additional testing as children with the syndrome are known to be at higher risk for a number of other conditions, including certain kinds of congenital heart defects and blood disorders. However, it is also important to remember that there will be support for your baby and your family in dealing with this kind of a diagnosis. Again, talk to your child's pediatrician for more information.