House lawmakers approved a comprehensive ban last month. The version approved Friday by the Senate would create a statewide law that would pre-empt local texting bans.
The Senate texting ban would allow the use of satellite navigation systems.
Forty-six states have laws against texting while driving. Texas lawmakers passed a statewide ban in 2011, but it was vetoed by then-Gov. Rick Perry.
The latest attempt drew new momentum after a March church bus crash killed 13 people. Federal investigators say the driver of the pickup truck that hit the bus said he was checking for a text when the crash happened.
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