ABOARD THE INDEPENDENCE, Lithuania—On the deck of this floating gas terminal, Mantas Bartuska awaits a tanker to pass a narrow inlet on the Baltic Sea with the first natural gas shipments from the Gulf Coast that many hope will transform Europe’s energy market.

“Soon, hopefully, U.S. gas will come,” said Mr. Bartuska, chief executive of the operator of the Independence, the gas terminal docked at the port city of Klaipeda, Lithuania.

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