DALLAS—In the race to build a bullet train in the U.S., a Texas group aims to be the first to market—to zip travelers from Dallas to Houston in 90 minutes as soon as 2021—without a dime in federal or state subsidies.

The $10 billion Texas Central Railway project touts itself as fully private, in contrast to a proposed, $68 billion high-speed train system in California that is supported by taxpayers and has been delayed by legal and political opposition.

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