I have a black horizontal line appearing on my iPad.

Start with doing a Reset:

Press and hold both the Sleep/Wake button and the Home button for at least ten seconds, until the Apple logo appears (ignore the Slide to Power Off option that shows up first).

Probably a hardware failure. Hope it's still under warranty.

You may want to try a reset. Hold the Sleep and Home button down for about 10 seconds until you see the Apple logo. Ignore the red slider.

Try resetting the device. Hold down the sleep and home key, past when you see the power down slider and until you see the silver apple. Let it reboot and see if it goes away.

If that doesn't do the trick you may need to do a restore to factory defaults adn restore from backup...but honestly I think it's something fundamentally wrong and your best bet is to take it to an apple store and have them look at it

Me too. It is a band of black and colored lines running vertical in landscape and horizontal in portrait about two inches wide. It's cover over all apps and all the time. Reboot, restore is no help but touch function works fine across the stripes. The screen capture from the iPad1 does not appear the band of black and colored lines. It shows a normal screen. The answer when asked at the Genius Bar is, they never seen this problem before. And the solution that fix the problem may be expensive. But when searching on Google with the keyword "iPad half screen black" it's return 64,800,000 results. Some user comment that the video display hardware on the system board has been damaged. It's possible something has been dislodged or come loose. But, again, why this happened to many many users in the same period? In most cases the iPad never drop. It's happened immediately with no sign of warning. There are no ways to fix it right now. The iPad1 is just 2 years released. Is it too early to replace with the iPad3?

It looks to me that the equivalent of the video card has gone bad. If it's still under warranty take it into your local apple store. If not....you may want to explore either the out of warranty repairs - might qualify for the out of warranty repair price structure, pay $269+ (Depends on model) to exchange your iPad for a refurbished one.