If Wii gives Disc Read Error, first step you need to check is to try another disc or clean the exisitng disc. This will confirm that problem is with either Wii or Disc. If the console itself is giving disc read error then it needs to be repaired. You may check out DIY Wii Fix Guide, that will show step by step information to fix the problem yourself. It also gives information on fixing many Wii Related Problems, so its a kind of Wii Problems Reference Fix Guide

Hmmm, I'd take it back and exchange it if I were you. As long as the label is to the right, it should be playing your discs. You can try to contact Nintendo at 1-800-255-3700