The two have been engaged since October, when West rented out AT&T Park, home of the San Francisco Giants, and proposed, presenting Kardashian with a diamond ring.
Grammy-winning West, 36 and Kardashian, star of "Keeping Up with the Kardashians," started dating in April 2012 and have a daughter, North West, who was born in June.
It has been widely speculated that the couple are planning a Paris wedding this year.
In a behind-the-scenes video posted on Vogue.com, the couple is seen laughing together and doting over their baby, who also makes an appearance in the photo spread.
It will be the third marriage for Kardashian. She wed her second husband, NBA basketball player Kris Humphries, in an elaborate ceremony filmed for the reality TV show, but the union lasted only 72 days. Her first marriage to music producer Damon Thomas also ended in divorce.
The April Vogue cover can be seen at AmericanVogue.com; the issue hits newsstands on March 31st.