Did Roman Catholic Church law ever condone (the contemporary practice of) incest? I may not be asking the right question, but I'm trying to figure out the context behind a quote from an Arthur Pink sermon:


  herein we may see the blasphemous impiety of the popes of Rome, who in the canons have dared to dispense with some of the laws of consanguinity in Leviticus 18


What's he talking about?

A little vocabulary help from Wikipedia:


  Consanguinity ("blood relation", from the Latin consanguinitas) is the property of being from the same kinship as another person. In that aspect, consanguinity is the quality of being descended from the same ancestor as another person
At the time Mr Pink (1886—1952, according to Wikipedia) was writing, this was decidedly not the case: not only were all these prohibitions included in canon law, but in fact canon law was a bit more restrictive, prohibiting marriage between first cousins as well.
The Church does currently allow marriage between a person and their sibling-in-law. This freedom is new since 1983, when the code of canon law was revised. Then again, though Leviticus doesn't permit such marriages, Deuteronomy does in at least one instance. None of the other prohibitions given in Leviticus 18 are, or apparently can be, overridden in the Church; nor, it seems, has this ever been possible.
Leviticus 18:7–18, with prohibitions extended to both sexes, forbids intercourse (and thus, in the eyes of the Church, marriage) with:

One's parents and step-parents (vv. 7–8)
Siblings, step-siblings, and half-siblings (vv. 9, 11)
Children and grandchildren (vv. 10, 17)
Aunts and uncles (vv. 12–14)
Children-in-law and siblings-in-law (vv. 15, 16, 18)

How does Church law respond, and how has it responded, to these prohibitions?
Canon law reckons consanguinity (blood relationship) and affinity (legal relationship by marriage) in lines and degrees.
The direct line of relationship is the direct parent-to-child (or parent-in-law to child-in-law) relationship. The collateral line of relationship involves siblings at some point.
The degree of relationship of two people is the sum of the numbers of people in the line to the last common ancestor, not counting the common ancestor. For example, my sister and I are related in the second degree of the collateral line (count 1 for her plus 1 for me; the parents—our common ancestors—aren't counted). My niece and I are related in the third degree (count one for me, one for my niece, and one for her father, my brother). And so on.
The current Code of Canon Law has the following in its discussion of impediments to marriage:

Canon 1091 §1 In the direct line of consanguinity marriage is invalid between all ancestors and descendants, both legitimate and natural.

This prohibits incestuous marriage between a person and their parents, children, or grandchildren (prohibitions 1 and 3).
Similarly, we have:

Canon 1091 §2 In the collateral line marriage is invalid up to and including the fourth degree.

This prohibits marriage between siblings, including half-, step-, or adopted siblings (prohibition 2); aunt/uncle and nephew/niece (prohibition 4); first cousins; and great-aunt or -uncle and great-nephew or -niece.
Although some of the Church's restrictions on marriage (for example the restriction that you must marry a Catholic) may be overridden, or dispensed, for what is typically called &quot;just cause&quot;, the restrictions on marrying parents, children, or siblings may not be dispensed:

Canon 1078 §3. A dispensation is never given from the impediment of consanguinity in the direct line or in the second degree of the collateral line.

Canon lawyer Cathy Caridi, in her excellent article &quot;Can Cousins Marry in the Church?&quot;, discusses the possibility of first cousins—relatives by consanguinity in the fourth degree—marrying (briefly, it is forbidden by canon law but for very significant reasons the bishop may dispense from the prohibition); she does not address the possibility of aunt/uncle and nephew/niece marrying (these are relatives in the third degree; the marriage is prohibited by canon law but it might be possible that a bishop would be able to dispense from the prohibition; however, I have not been able to find any instances in which this prohibition has been dispensed).
The remaining prohibition is number 5, prohibition from marrying one's children- or siblings-in-law. Importantly, the Old Testament itself makes an exception for marrying one's brother-in-law in certain cases:

When brothers live together and one of them dies without a son, the widow of the deceased shall not marry anyone outside the family; but her husband's brother shall go to her and perform the duty of a brother-in-law by marrying her.

(Deuteronomy 25:5)
The older code of canon law, published in 1917 (and based on a reading of the many collections of writings which had constituted Church law until that point) included all the prohibitions discussed above, and did in fact prohibit marriage to either children-in-law or siblings-in-law:

Canon 1077 §1 Affinitas in linea recta dirimit matrimonium in quolibet gradu; in linea collaterali usque ad secundum gradum inclusive.
[Affinity in the direct line invalidates marriage in any degree whatsoever; in the collateral line up to and including the second degree.]

This prohibits marriage to parent- or child-in-law (affinity in the direct line) as well as to sibling-in-law (the collateral line in the second degree).
This code of canon law was completely replaced in 1983, and all canons in it nullified unless they were replaced by equivalent canons in the current code. In the current code, the prohibition of marriage to parent- or child-in-law is still in place:

Canon 1092. Affinity in the direct line in any degree invalidates a marriage.

But (especially, it appears, in consideration of the passage from Deuteronomy) the prohibition of marriage to sibling-in-law is not considered an absolute to which one must hold, and thus this sort of marriage is no longer prohibited; but it, and all the other prohibited marriages listed in Leviticus 18, were indeed prohibited at the time of Mr. Pink's sermon.