Objection 1. It would seem that an illegitimate son cannot be legitimized. For the legitimate child is as far removed from the illegitimate as the illegitimate from the legitimate. But a legitimate child is never made illegitimate. Neither, therefore, is an illegitimate child ever made legitimate.
Objection 2. Further, illegitimate intercourse begets an illegitimate child. But illegitimate intercourse never becomes legitimate. Neither, therefore, can an illegitimate son become legitimate.
On the contrary, What is done by the law can be undone by the law. Now the illegitimacy of children is an effect of positive law. Therefore an illegitimate child can be legitimized by one who has legal authority.
I answer that, An illegitimate child can be legitimized, not so that he be born of a legitimate intercourse, because this intercourse is a thing of the past and can never be legitimized from the moment that it was once illegitimate. But the child is said to be legitimized, in so far as the losses which an illegitimate child ought to incur are withdrawn by the authority of the law.
There are six ways of becoming legitimate: two according to the canons (Cap. Conquestus; Cap. Tanta), namely when a man marries the woman of whom he has an unlawful child (if it were not a case of adultery), and by special indulgence and dispensation of the lord Pope. The other four ways are according to the laws: (1) If the father offer his natural son to the emperor's court, for by this very fact the son is legitimate on account of the reputation of the court; (2) if the father designate him in his will as his legitimate heir, and the son afterwards offer the will to the emperor; (3) if there be no legitimate son and the son himself offer himself to the emperor (4) if the father designate him as legitimate in a public document or in a document signed by three witnesses, without calling him natural.
Reply to Objection 1. A favor may be bestowed on a person without injustice, but a person cannot be damnified except for a fault. Hence an illegitimate child can be legitimized rather than "vice versa"; for although a legitimate son is sometimes deprived of his inheritance on account of his fault, he is not said to be illegitimate, because he was legitimately begotten.
Reply to Objection 2. Illegitimate intercourse has an inherent inseparable defect whereby it is opposed to the law: and consequently it cannot be legitimized. Nor is there any comparison with an illegitimate child who has no such defect.
Source: Summa Theologica (New Advent)