After i got my xbox replaced i found out i had to re-install all the games on my HDD and one of the games i had installed was Mass Effect.  I deleted the game from my HDD and tried to re-install it but everytime i do i keep getting a disc read error. 
 
I cleaned my disc and tried it again but it still won't let me install it.  I did play the game from the disc and loaded up one of my saves so i'm not sure it's a problem with the HDD or my disc is just damaged beyond repair. 
 
 Has this ever happened to anyone? 

This happened to my friend with a copy of Halo 3 that he severely scratched. I suggest getting a loan of a copy and installing that so then you can use your own.

Hmm, i hadn't played ME that recently so i didn't think was to scratched up but maybe i'll see if i can get a replacement disc if that's the problem. 
 

I've had this happen to a few games I've gotten from GameFly. They might have the occasional freeze and I will decide to just install it to my HDD, and countless of times, the installation will simply fail. Each game has a certain percentage it would have stopped at. I did this on my own with Fight Night Round 4 by accident, by banging on my desk hard enough to cause my X360 to start making loud noises.
 
Lesson(s) learned: Install the games you rent online before you start playing them, and don't bang on your desk and destroy the discs for other gamers.

Once it's installed you just need the disk in to run it.
If a friend's got it, borrow theirs, install it, give it them back.

happened to me too.   I think it's that my new 360's disc drive is crap.   So I bought it on steam.   Gotta have my Mass Effect.