Ulster came up short by four points against Bordeaux in their final European Champions Cup game of the season.
Former Leinster out half Ian Madigan opened the scoring with a penalty in front of the posts after two minutes when Ulster went off their feet at the ruck.
Bordeaux got the game's opening try on 12 minutes, Ulster had a scrum on their own 22, the French side got a powerful drive on, Ulster scrumhalf Dave Shanahan was turned over by Yoan Lesgourgues and the visiting pivot dashed over unopposed with Madigan converting.
Ulster struck back on 18 minutes with a try from captain Andrew Trimble.
Les Kiss' side kicked a penalty to the corner, from the lineout the forwards made hard yards before Iain Henderson drove for the line, the Irish flanker was stopped short, Shanahan switched the attack to the blind side by feeding Paddy Jackson, the out half put Trimble over in the corner but Jackson failed to convert.
Bordeaux's powerful scrum paved the way for their second try on 23 minutes.
Ulster had a scrum in their own 22 but were driven backwards and as they retreated over their own line No. 8 Marco Tauleigne picked up and scored with Madigan converting.
Ulster responded with a try five minutes later.
They kicked a penalty to touch, from the lineout the ball was moved in field, centre Stuart McCloskey had a powerful run but was held up; the ball was moved wide to Tommy Bowe, the ball was quickly recycled and lock Franco van der Merwe dived over from close range and Jackson added the conversion.
Madigan added a long range penalty for the visitors three minutes later.
Ulster nearly got their third try in the final minute of the first half, Trimble crashed through midfield before releasing to Jackson, the Irish out half showed great pace but was stopped close to the line however Bordeaux fullback Geoffrey Cross was yellow carded for slowing the ball down.
Jackson's penalty reduced Bordeaux's lead to 20-15 at the interval.
Jackson was off target with a 54th minute penalty after Bordeaux infringed at the ruck.
Ulster struck the front for the first time on 58 minutes with their third try.
They kicked a penalty to the corner, Clive Ross took the lineout, Stuart McCloskey had a probing run for the line but was stopped short, Paul Marshall spun the ball to Darren Cave and the centre held off two defenders to score right on the post pad with Jackson converting.
Bordeaux went back in front on 70 minutes with a penalty from replacement Simon Hickey.
Hickey added another penalty four minutes later which sealed the game for the Top 14 side.
Ulster had a chance to win the game with the final play, they kicked a penalty to touch, from the lineout Henderson drove for the line but he was held up in the tackle.
C Piutau, T Bowe, D Cave, S McCloskey, A Trimble, P Jackson, D Shanahan,
A Warwick, R Best, J Simpson, R Diack, F van der Merwe, I Henderson, C Henry, S Reidy
J Andrew, C Black for Warwick 71 mins, R Lutton for Simpson 24 mins, K Treadwell for Diack 46 mins , C Ross for van der Merwe 55 mins, P Marshall for Shanahan 40 mins, B Herron, C Gilroy for Bowe 63 mins
G Cros, N Ducuing, J-B Dubie, J Wakacegu, B Connor; I Madigan, Y Lesgourgues,
S Kitshoff, C Maynadier, M Clerc, T Palmer, B Botha, L Madaule, L Goujon, M Tauleigne
B Auzqui for Maynadier, S Taofifenua for Kitshoff 37 mins, V Cobilas for Clerc 55 mins, J Edwards for Botha 41 mins, H Chalmers for Taulegine, J Audy for Lesgourgues 51 mins, S Hickey for Madigan 59 mins, J Rey
Ref M Carley RFU