South Korea's Ministry of National Defense has issued a statement calling on Pyongyang to stop making what it calls unacceptable threats.
The ministry also warned Saturday that the South's forces are in full readiness posture and will completely punish the North if there is any provocation.
In a statement carried by the North's official KCNA news agency Saturday, Pyongyang said it has entered a "state of war" against its southern neighbor and "all issues raised between the North and the South will be handled accordingly."
A number of North Korean web sites were apparently targeted by hackers after that announcement, and have been intermittently unavailable Saturday. While no group has claimed responsibility for the distributed denial of service attacks, several individual hackers have told VOA they are responsible for the outages.
Later on Saturday, Pyongyang threatened to shut down a joint industrial complex with South Korea.
A spokesman for the North's office controlling the Kaesong industrial complex said it would close the facility just north of the Demilitarized Zone separating the two countries if Seoul continues to continues to undermine the North's dignity.
Seoul's Unification Ministry in charge of North-South relations in lieu of diplomatic ties says the war decree is not new, but a continuation of provocative threats.
However, a spokeswoman for the National Security Council in Washington says U.S. officials are taking Pyongyang's announcement seriously.
The North's young leader, Kim Jong Un, who succeeded his father, Kim Jong Il after his death in December 2011, declared on Friday that his forces are ready "to settle accounts" with the United States.
That remark came after a pair of stealth B-2 bombers were dispatched from the United States to drop dummy bombs on a South Korean island range.
The mission, following those by B-52 bombers earlier in the month, was seen as sending a message to both Seoul and Pyongyang. It was a dramatic reassurance to the South that it is protected under the U.S. nuclear umbrella, and a warning to the North of the U.S. capability to strike it precisely and quickly from a long distance, should war erupt.