Anne Mørkve from Eldar IL contested her first race in the Biathlon European Cup for Juniors in Geilo in 2004 . The highlight of their first season was the 2005 Junior World Championships in Kontiolahti . In the individual, she just missed a medal in fourth place against Marie Dorin and was 23rd in the sprint and 17th in the pursuit. Two years later in Martell the results were eleven in the individual, 45 in the sprint and 36 in the pursuit. Since the beginning of the 2007/08 season Mørkve started regularly with the women in the European Cup and the later IBU Cup . In her first sprint in Geilo, the Norwegian was 39th, in the following sprint she not only won first points, but also achieved a top ten position for the first time in eighth place. In Osrblie in 2008 she also joined the Germans Tina Bachmann , Romy Beer and Carolin Hennecke second in a German-Norwegian mixed relay.
Nationally Mørkve won the silver medal in the relay race of the Norwegian Championships in 2007 in Folldal alongside Jori Mørkve and Liv-Kjersti Eikeland, representing the Hordaland region . In 2010 she won her first championship title with Birgitte Røksund and Jori Mørkve with the relay.