The ban would be largely symbolic given the scale of the current military offensive against the LTTE.
An original ban on the LTTE was lifted in September 2002 ahead of an Oslo-backed peace process, which collapsed when President Rajapakse pulled out of the truce last January.
The Tigers have not formally responded to the government's ultimatum.
The group, which is already outlawed by the United States and the European Union, has been waging a drawn-out campaign for independence for Sri Lanka's minority Tamil community.
The fall of Paranthan, just outside the Tigers' political capital of Kilinochchi, has effectively cut off the main supply line to several other vital rebel strongholds, the ministry said in a statement.
At least 50 Liberation Tigers of Tamil Eelam (LTTE) rebels were killed and twice as many wounded in heavy fighting, the ministry said, without stating whether security forces suffered any casualties.
"Unable to withstand the fury of the combined army and air