Actually the LDS church temporarily abandoned polygamy around 1904 and really started enforcing this ban in 1906 with purges in 1907 that resulted in excommunication and disfellowshipment outcomes in church courts for anyone entering post-1904 polygamous relations. The LDS church issued a Manifesto in 1890 to temporarily stop new polygamous marriages in Utah so they could get statehood. During the Reed Smoot hearings it was exposed that there were still some polygamy marriage happening, including one in 1896 when Wilford Woodruff took another plural wife. Polygamous marriages were being performed on the High Seas, Mexico and out-of-state in small numbers with great secrecy. However, as far as anyone can tell, the LDS First Presidency has been very strong in enforcing the Polygamy ban since 1904.
This temporary ban is still in-force but all honest church members acknowledge that it's just a temporary ban. If Polygamy can be legalized then the LDS church will probably resume it's practice. See http://scriptures.lds.org/dc/132 for the text of the church's doctrine on polygamy.