Hearts 0-2 Bordeaux (Aggregate: 1-2)
Hearts bowed out of the Uefa Cup as Bordeaux booked their third round place with two well-taken goals.
The home side's first-leg lead was quickly cancelled out by Albert Riera's 25-yard effort.
The visitors, who hit the woodwork twice before the interval, then doubled their lead when Pascal Feindouno found the back of the net midway through the second half.
Roared on by the Tynecastle crowd, Hearts did create several opportunities but were unable to find their way past a well-drilled Bordeaux defence.
Hearts served early notice of their intentions when Jean-Louis Valois let fly from outside the area following a fifth-minute short corner from Paul Hartley.
8 mins: Riera scores with a stunning long-range shot
66 mins: Feindouno fires into the top corner
But the tie was level on aggregate in the eighth minute when Riera fired past Craig Gordon after Valois had gifted possession to Jean-Claude Darcheville.
Patrick Kisnorbo also tried his luck from distance for Hearts while Phil Stamp's effort was parried by Ulrich Rame.
At the other end, former Nottingham Forest striker Darcheville volleyed against the bar from the edge of the area.
After Hearts midfielder Hartley had been stretchered off with a facial injury, Bordeaux rattled the woodwork again when Gordon pushed Feindouno's shot onto his right-hand post.
As half-time approached, Hearts were denied twice by the officials.
Referee Grzegorz Gilewski waved away a penalty appeal after Franck Jurietti appeared to handle.
Then, a minute later, Stamp had the ball in the net only for the goal to be ruled out for offside.
Hearts keeper Craig Gordon cannot stop Bordeaux taking the lead
Tempers began to flare after the break.
But with 25 minutes left, Feindouno brought the game to life with a superb shot into the top corner from just inside the box.
That left the home side needing to score twice.
Stamp blazed over the bar and Rame beat out a Stephen Simmons effort before Hearts boss Craig Levein made his final change, throwing Andy Kirk into the attack in place of Valois.
However, it was not to be the Edinburgh side's night - as Bordeaux finished as worthy winners.
Hearts: Gordon, Maybury, McKenna, Webster, Neilson, Stamp, MacFarlane (Simmons 45), Kisnorbo, Valois (Kirk 73), de Vries, Hartley (Severin 30).
Subs Not Used: Moilanen, McCann, Wyness, Sloan.
Booked: Simmons, de Vries, Kisnorbo, McKenna.
Bordeaux: Rame, Caneira, Alicarte, Pochettino, Jurietti, Costa, Darcheville (Celades 82), Riera (Da Costa 45), Feindouno, Planus, Chamakh (Sahnoun 90).
Subs Not Used: Roux, Deivid, Basto, Miranda.
Booked: Caneira, Rame.
Referee: Grzegorz Gilewski (Poland).