Joseph signed a four-year deal with the Broncos, according to ESPN's Adam Schefter. Financial terms were not disclosed.
Joseph interviewed for nearly four hours with Denver on Tuesday and was scheduled to meet with representatives from the San Diego Chargers the following day. However, the 44-year-old Joseph elected to return to the Broncos' Englewood facility, the Denver Post reported.
Joseph was the last of three candidates to interview for the Broncos. Denver also interviewed Kansas City Chiefs special teams coordinator Dave Toub and Atlanta Falcons offensive coordinator Kyle Shanahan last weekend while their respective teams were on a bye.
Joseph played a significant role in his lone season with Miami, turning around a Dolphins' team that finished 2015 with a 6-10 mark by helping the club to 10 wins and the franchise's first postseason berth since 2008.
Kubiak, who was plagued by a complex migraine condition, announced his retirement from coaching following a season-ending 24-6 victory over Oakland.