Sri Lankan naval craft shelled the first ship after it failed to stop for an inspection and opened fire on the navy, ministry spokesman Prasad Samarasinghe said.
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He said a second vessel in the area was also challenged and it too had opened fire at the navy, prompting a sea battle.
"Both vessels were attacked and they exploded and sank," Samarasinghe said.
The defence ministry in a statement said the vessels had no flags or other identification markings.
After firing warning shots across the bows the navy shelled the ships, the ministry said, adding the sea battle occurred in deep water off the island's Arugam Bay coast.
"The naval ships observed huge explosions onboard the vessel, confirming that it was carrying explosives and war-like material to the LTTE (Liberation Tigers of Tamil Eelam)," the ministry said immediately after the first vessel was said to have been sunk.
It said the second vessel retaliated with gunfire just after noon (0700 GMT).
"Naval gunfire