957/8. Archbishop Oda divorces Eadwig from Ælfgifu
One version of the Anglo-Saxon Chronicle notes that Archbishop Oda separated King Eadwig and Ælfgifu, because they were too closely related. (The date given is 958, but since the previous annal, dated "957", gives events of 956, and the following annal, dated "959", gives events of 959, it is not clear whether the divorce actually took place in 957 or 958.) Since the relationship between them was most likely known when they were married (and indeed, one would expect that Oda of Canterbury performed the marriage), it seems more likely that this divorce stemmed from political reasons than religious ones. It may even be that Eadwig and Ælfgifu were not in fact too closely related by the church definition.
There is no other information on the kinship of Eadwig and Ælfgifu in Eadwig's reign. In Edgar's reign, a woman called Ælfgifu who is described as a matrona (i.e., a married woman) and a kinswoman of Edgar (as Eadwig's queen would be, by marriage) receives two estates in 966 (S 737 and 738), and her will also survives, though unusually it mentions no living husband or descendents (S 1484). If it is assumed that this Ælfgifu was Eadwig's queen, then we can determine precisely how closely related Eadwig and Ælfgifu were, because the Ælfgifu of the will was the sister of Æthelweard the Chronicler, who records that he was the great-grandson of King Æthelred (865-71). This would make Ælfgifu the great-granddaughter of Æthelred, and since Eadwig was the great-grandson of Æthelred's brother Alfred, the two would fall within eight "degrees of propinquity".
This gets a bit technical. The old Roman system recognized a cognatio, or kindred, as people within seven "degrees of propinquity". To find out the number of degrees, one counts up to the nearest common ancestor and then back down again. Pope Gregory II, in the 721 Synod of Rome, stated simply that marriage was forbidden within the cognatio, or within seven degrees of propinquity: on which basis Eadwig and Ælfgifu would be perfectly legitimate.
The Germanic system, in contrast, counted the number of generations from the common ancestor: on that basis, Ælfgifu and Eadwig, sharing a great-great-grandfather, would be in the fourth generation. Synods in the second half of the 8th century (e.g. Compi?gne and Verberie) translate Gregory's ruling by declaring marriage null and void within the first three generations.
The crux comes with the first Anglo-Saxon legislation on the subject, in a code of Æthelred from 1008, which states that marriage is forbidden "within six degrees, that is to say, four generations" (VI Æthelred 12). This is not in fact the same thing: six degrees would be three generations. The distinction only matters to people within the fourth generation, but unfortunately for them, Eadwig and Ælfgifu fell into that category. If a similar law had been in place in the 950s when Eadwig and Ælfgifu got married, one might imagine that they were given the benefit of the doubt as not being within six degrees. And perhaps it was only when Eadwig began making changes and antagonizing people (q.v.) that people remembered, and chose to apply, the stricter interpretation of the letter of the law.
M. Deanesly and P. Grosjean, "The Canterbury Edition of the Answers of pope Gregory I to St Augustine", The Journal of Ecclesiastical History 10 (1959), pp.1-49
N. Brooks, The Early History of the Church of Canterbury (Leicester: 1984), pp.225-6