Blasts powerful enough to be heard in downtown Donetsk came from the area near the airport, which has been under the control of government troops since May and has come under unremitting attacks from pro-Russia separatist rebels since then.
The cease-fire had appeared to be holding for much of the day on Saturday, but shelling started late at night. A rebel statement said Ukrainian forces violated the cease-fire by firing on their positions in six locations on Saturday, including near the Donetsk airport. The statement said several rebels were killed.
Witnesses told The Associated Press that heavy explosions were also coming from the eastern outskirts of the port city of Mariupol late Saturday, where Ukrainian troops retain defensive lines against the rebels.
The city council there said one serviceman was wounded when a checkpoint was hit and the volunteer Azov Battalion said on Facebook that their positions were hit by Grad rockets, but did not give details.
Mariupol is located on the coast of the Sea of Azov, 70 miles south of Donetsk. Rebels recently opened a new front on the coast, leading to fears that the separatists were trying to secure a land corridor between Russia and the Crimean Peninsula, which Russia annexed in March.
Ukraine, Russia and the Kremlin-backed separatists signed the cease-fire deal in the Belarusian capital Minsk on Friday in an effort to end more than four months of bloodshed in the region.
The negotiators also agreed on the withdrawal of all heavy weaponry, the release of all prisoners and the delivery of humanitarian aid to devastated cities in eastern Ukraine.
Western leaders voiced scepticism over Russia's commitment to the deal. A previous 10-day cease-fire, which each side repeatedly accused the other of violating, yielded few results at the negotiating table.
Ukrainian President Petro Poroshenko's office on Saturday said he and Russian President Vladimir Putin had discussed steps for giving the cease-fire a stable character in a telephone conversation.
But, it said, both leaders assessed the cease-fire as having been fulfilled as a whole.
Fighting between pro-Russian rebels and Ukrainian government troops has ravaged the already teetering Ukrainian economy, claimed at least 2,600 civilian lives and left hundreds of thousands homeless, according to United Nations estimates.
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