The move follows, among other things, a December pileup that shut down a Seattle highway for more than an hour. Police blame a driver who was using his BlackBerry.
The measure now goes to Washington's Senate.
If it becomes law, Washington would be the first state to bar DWT.
At least three others are considering it, according to The Early Show consumer correspondent Susan Koeppen.
A wireless industry trade group, CTIA, The Wireless Association, calls focusing on texting-while-driving too narrow.
There are lots of things that distract us when we drive, the group points out, such as eating and grooming so, while the CTIA agrees it's not wise to text while driving, it says lawmakers should tackle driving distractions on a wider scale.
In November, Koeppen reported on a .
She says Patrick Sims spent 10 days in jail, is still on probation, and isn't allowed to text-message.