Sit with a lyrics sheet and listen to the song a few times, reading the lyrics as you go. Then, replay the song with the lyrics in front of you and sing the lyrics with the song. If you can move on to singing the lyrics with the song playing but without the lyrics sheet then do so. If you still have trouble then sing the lyrics as you read them from the sheet.
Turn off the music and break the lyrics into major stanzas and chorus. Go at the chorus first by seeing if you can rattle it off without stumbling. If you can't, memorize the lyrics of the chorus line by line. After you get each line, write it out (it's said that longhand works better). Do the same with the lyrics for the non-chorus stanzas until you can comfortably recite each of the lines without messing up.
Fill your idle time with recitations of the lyric in full from the beginning of the song to the end. Since you need to learn the lyrics quickly, using the five minutes you waste standing in line or the 15 minutes of driving to work will quickly add up to lots of memorization time that you otherwise wouldn't have.
Sing the song all the way through without any recording to back you up. If you mess up a lyric, or even just falter without actually getting the lyrics wrong, make a note of where you're having trouble. Go back to the parts of the song that you studied before and re-study the entire stanza that contains the word or line you messed up on.
Take the pressure off by finding other words that match the song well enough for you to get by. This will help you sing the song naturally so you can concentrate on a good performance after having learned the lyrics to the best of your ability.