MOSCOW—President Vladimir Putin ordered an increase in Russian spying on U.S. development of short- and intermediate-range missiles and said Moscow would match any the U.S. builds, after the collapse of a treaty that outlawed the weapons.

The statement was the furthest-reaching response from the Kremlin to the end of the Cold War-era Intermediate and Medium-range Nuclear Forces Treaty between Washington and Moscow. Washington withdrew from the treaty last week, letting it lapse after accusing Moscow of violating the pact.

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