Can I enter U.S. right before my visa expires?

My B1/B2 visa expires in April 2007. Can I enter the country say in late March 2007, get granted six months (presumably) stay on newly issued I-94 card and consequently be in the country till say September 2007? In effect, I would be in the country while my pport visa expired, but my I-94 permits me to stay. Is this possible and common?