The broadcaster currently shares the contests with Sky following a protracted disagreement between the two firms over who had the broadcast rights.
BT’s new deal starts from the 2018/19 season and will see it screen up to 134 matches per season from both competitions, with the firm promising its coverage will include at least 22 clubs from England, Wales, Scotland and Ireland.
Andy Haworth, BT’s managing director of content and strategy, said: “BT Sport is now the home of elite European rugby, as well as the home of Aviva Premiership Rugby.
“We’re proud to have won all of the live rights for this fantastic competition. We’re extremely passionate about our rugby coverage and will continue to work hard to grow the profile of the sport.”
Vincent Gaillard, CEO of European Professional Club Rugby, said “We are delighted to move to a single pay-TV platform for European club rugby with a premium broadcaster in BT Sport, who share our commitment to the promotion of our competitions.
“We look forward to further developing that relationship to continually create outstanding rugby experiences for fans across the United Kingdom and Ireland.”