Sebastien, try calibrating the battery before you get too worried about anything. If your old battery was in bad shape, then the computer may still be working off the data it knows from that old information. In order to reset that data, go through the calibration procedure of basically charging the battery to 100% (or as high as it'll go) then using it until it dies completely. Do that like three times and that should allow the computer to relearn the battery's charge characteristics as they are now, rather than as they were with the old battery.
Here's a little more information on the subject.
[[Apple Laptop Battery Troubleshooting#Section_Battery_Calibration|Apple Laptop Battery Troubleshooting - iFixit|new_window=true]]
[[Battery Calibration|Battery Calibration - iFixit|new_window=true]]
Give that a try; I did the same thing on the MacBook Air my son gave me that had a bad battery and it reset everything after about three cycles and worked perfectly after that. If it's still not charging fully after that, come on back and we'll see what else we can figure out.