SHORT ANSWER- My great-great grandparents were first cousins.
LONG EXPLANATION. It has taken me 20 years to figure out this complicated story, and it has taken me 2 days to figure out how to explain it to you. . .
Yes, some people marry their first cousins. But the Roman Catholic Church forbids marriage between relatives of "four degrees of consanguinity" (people who share great-great-grandparents, or anyone more closely related). Such a marriage may be permitted, however, if the couple has received a dispensation from a church official.
When Constant Cartier and his mother's first cousin Marie Louise Brousseau married in 1803, they had received such a dispensation. (Thank you, Jonathan Boyer, for finding and translating this record!) Present at this wedding was Constant's brother, Augustine Cartier.
Forty-two years later, Augustine and his family were living in the tiny hamlet of Cooperville, New York, about 28 miles from their hometown of L'Acadie, Quebec. Also living in Cooperville was his sister Josephte Cartier, her husband Theodore Bechard, and their family.
On 28 January 1845, both sets of parents were present when Augustine's son Constant, married his first cousin, Florence Bechard, Josephte's daughter. But there was no dispensation. Why? Did Augustine and Josephte forget that their own brother had needed a dispensation to marry their mother's cousin? Or did they think that they could get by without a dispensation if the priest did not know about that relationship ? Or was the priest himself unfamiliar with the church rules about consanguinity?
The following November, their son Louis was baptized in Cooperville. The record indicates that he was born of "the legitimate marriage" of his parents.
But things got interesting when the family moved back to L'Acadie. Here the priest knew their family, and informed them that their marriage was invalid. The baptismal records of sons Philippe (1847) and Joseph (1849) do not include the standard phrase "legitimate marriage." Constant and Florence must have begun a complicated process which took several years, but finally enabled them to be legally married in the eyes of the Catholic Church.
Their unusual marriage record, dated 14 March 1851, mentions "the dispensation of second degree relationship," and recognizes their three children as legitimate, but does not include the customary mention of the couple's parents or residences. But finally Constant and Florence were home in Canada and legally married. They went on to have six more children, including my great-grandmother Melanie.