If the marriage is annulled, then they may receive communion.
This is often a contentious topic as many catholics and non-catholics do not understand the annullment process. Suffice it to say that catholics are no different from their non-catholic brethren. Sometimes, they make mistakes and marry for the wrong reasons. These issues are evaluated by the Church during an annullment inquiry. If it is determined that the couple were not in a 'valid' catholic marriage (one of the two chose not to have children and prevented procreation; the couple never attended pre-Cana classes, etc), then the 'marriage' is annulled, since it wasn't a marriage by catholic understanding, in the first place.
I notice, that Ted Kennedy made a mistake. Is there no deadline after which you have to live with your mistake?