Determining the outsider in "Desiree's Baby" depends on ones point-of-view. We will look at a few different possibilities depending on different personal interpretations of the question.
Desiree: Desiree could be considered an outsider in the story. First of all, she is adopted. She does not have a true name until she is given one by Armand. Second, she is ostracized by Armand at the end of the story. She is told by Armand that she is not white and that she is no longer a part of his family- basically, she has disgraced his name and can no longer be a part of his life.
Armand: Armand can be considered an outsider given his family history. His mother admits in a letter to his father that he is part black. Given this information, Armand would be ostracized by white society because of his mixed background. He is not completely white and, therefore, cannot be considered as a part of high society.
Desiree's baby: The child of Armand and Desiree can be considered an outsider given it has the blood of both white and black parents. Children, during this period, needed to be "pure" to be given the rights of white society. This child had blood from a white mother and a partially black father. Given it was born into a "white family", discovery of its mixed heritage would ostracize the child.