It's classical music. The closing credits have a nice long chunk.
We hear bells as if from far away. Then a string section rises sadly into a melodic interplay with a horn section. After that dialog completes, it's back to the distant bells.
Not too many Americans have seen this interesting thriller because it's French. It does seem likely it will soon be remade by Hollywood (because of the way lip-reading is tied into the plot). However the soundrack might not be the same.
The composer's name is Alexandre Desplat, he'd been sort of France's Danny Elfman. When Sur mes lèvres came out he had already done scores for 77 movies&TV shows.
Other people I guess were impressed because it was just after that Desplat started getting hired by Hollywood. In the last few years his scores have included Girl with a Pearl Earring, The Upside of Anger, Syriana and Firewall (now in a theater near you).