The match marks the climax of the club's summer tour of 2013 which kicks-off with the fixture against England at Twickenham on 26 May.
Young will be in charge of the Barbarians' squad for the full tour and will have with him as manager the former Wales and Lions' forward Derek Quinnell. Between them Young and Quinnell went on six Lions' tours as players and both played for the Barbarians against international opposition.
Derek Quinnell said: " On behalf of my colleagues on the Barbarian committee it was a pleasure for me to invite David Young to be our coach again and we're delighted that he accepted the opportunity with such good grace.
"I have worked with him before - though it seems a long time ago that I was in the Wales' coaching set-up with Tony Gray and we sent for Dai to join our squad for the quarter-final of the first Rugby World Cup in 1987. He was a young 19-year-old prop at the time but was equal to the task and two years later he was back in Australia with the Lions and an automatic choice for the test team that won the series.
"The other great thing about Dai is that as a player and now an experienced and successful coach he understands what makes the Barbarians tick and fully respects and furthers the traditions of our club."
David Young, who coached the Barbarians to victories over England and Wales in 2011, said: "I have been privileged to coach the Barbarians on three previous summer tours in recent years but never mind how many times I am invited it is always an honour and one that I cherish.
"It is also a big responsibility to uphold the traditions of the Barbarians and to get the best out of the fantastic group of players they invariably assemble for these matches.
"This will be even more so next summer. The annual game against England is a big enough occasion it itself but to play the Lions on the first match of their own major tour is really a case of the icing on the cake. It promises to be a tremendous tour and all those of us involved will be committed to making it a big success."
The Barbarians and Lions played each other once before - at Twickenham in 1977 in honour of the Queen's Silver Jubilee. In the Lions XV that day were Derek Quinnell and his opposite number next year in Hong Kong in his role as manager of the Lions, the Scotland full-back Andy Irvine.
On that occasion the Lions defeated the Barbarians 23-14 in front of a capacity 60,000 crowd. Next June in Hong Kong the return match promises to be another glamorous and memorable occasion.
Barbarians Summer Tour 2013
Sunday, 26 May v England, Twickenham
Saturday, 1 June v British & Irish Lions, Hong Kong