Media giant Viacom may be suing YouTube in a $1-billion copyright-infringement suit, but it’s also taking a lesson from the online video service.

Following up yesterday’s item that Jon Stewart has extended his contract to host The Daily Show until the end of 2010, Viacom has unveiled a new website that will offer every minute of The Daily Show since it began in 1999 — and for free.

That’s 13,000 video clips in all, so the most avid fans of the satirical news show are unlikely to leave their computers for a very long time.

As The Los Angeles Times business section notes: “The site (www.thedailyshow.com) is meant to pull in advertising money from Day One, but it also will be something of a test lab for Viacom and perhaps for rivals looking over its shoulder.”

Viacom’s designers have been experimenting with ads that appear for two or three seconds at the start of a clip, recede, then emerge briefly from a corner of the picture like a network-TV promo while the video continues playing.

“Nobody wins when you have a 30-second ad in front of a 45-second piece of video,” says Erik Flannigan, executive vice president for digital media at MTV Networks, the Viacom unit that includes Comedy Central.