I am a lawyer in CT and practice in this area of the law.  The chances of a court giving her custody is slim as the court is going to determine the custody issue as to what is in the child's best interest.  if the ex does not have a job or positive/healthy environment to care for the child then the court is not going to let her have custody.  however, th court may permit the ex to have certain visitation rights.  However, i suggest that these be supervised and you should have the guy hire a lawyer to represent him in this to further show how he is the better parent - i.e. seeing that the guy has a lawyer says a lot.