Practicing slowly and increasing tempo gradually is definitely one good way to work on it. But since your problem isn't that the line is too fast per se, but rather that you can't maintain that tempo for the duration of the song, here's something else you might try: Play at tempo, but only for (say) four or eight bars, and then drop out for a couple/few bars (or just play a whole note for each of those bars). Then play another set of bars, and drop out again, etc. Over time, gradually increase the number of bars you play in a row, and gradually decrease the number of "rest" bars in between. This method has the same goal of increasing stamina, but has the advantage that you're always practicing at tempo (and can play along with the recording if you wish without having to slow it down).
Either way, keep in mind that the goal is ensure that your practice is always "perfect practice," as someone remarked previously. Every time you play a line sloppily you are practicing how to play sloppily. The idea of the gradual change across time, whether in speed or in number-of-bars-between-rests, is to make sure that when you are playing you are always correctly.