There are 2 artists with this name.
1. Babygrinder, hailing from Culemborg, the Netherlands, started back in 2001 under the name of Death by Disease. After a demo was finished the name was changed to Babygrinder, which by now consisted of 2 members: Chiram on guitars, and Astathica on drums, bass and vocals. Somewhere further along 2001 the "Mass Mutilation" demo was released.
In 2002 the first full-length, "Fetus Dinner" was released. At this time, Astathica did all the instruments himself. The album didn't contain any regular guitars, only bass, drums and vocals.
After a good 2-year break, the "Principle of Easily Skinned Flesh" (which indeed is a name pun on the similarly titled CoF album) EP was released. This contained a full sound, which was more grind-like. All instruments were once again done by Astathica.
Now, after a good 4 years of doing nothing, or not much, Babygrinder is back with an upcoming album. Songs that have been around in the underground, like "Wervelstorm" and re-recorded versions of "Born, cooked and eaten" and more are being mixed and the (still nameless) upcoming album's about to be done somewhere along 2008. Be sure to check www.myspace.com/grindingbabies for more info.
2. A jokingly awesome band from Australia. See http://www.myspace.com/babygrinderband for more info.