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1.
Among the only opportunities a hard-time prisoner would have to engage in sexual relations or intimate visitation with one of the opposite sex (usually his/her spouse). Does not always have a sexual connotation, but in this case, it almost always does.

Of or relating to the marital relationship or to the relationship between a wife and husband.
"No more conjugal visits until we find out who killed Percy"
by Mr C. December 11, 2003
 
2.
Visiting your partner who is currently in prison to have sex with them. Conjugal visits often motivate inmates to maintain good behavior.
Steph: God damn Bernie Madoff is a spunk!

Ashley: Yeah, I'd pay him a conjugal visit!
by Pepperonilawl September 27, 2009