Pistorius’ defence team was served with the notice on Friday, and the High Court in Pretoria and the Supreme Court of Appeal were sent the notices on Monday, spokesman Velekhaya Mgobhozi said.
Mgobhozi said the State would wait for transcripts of the court’s record before filing its papers.
In December, Judge Thokozile Masipa granted the State leave to appeal Pistorius’ culpable homicide conviction, but not his five-year jail term.
Pistorius will continue serving his current sentence in the meanwhile.
On October 21, Masipa sentenced Pistorius to five years in jail for the culpable homicide of Steenkamp. He may apply to be released on correctional supervision after serving a sixth (10 months) of the term.
The State filed papers in November calling for a heavier conviction and harsher sentence.
Prosecutor Gerrie Nel argued that the sentence imposed on Pistorius was “shockingly inappropriate”. However, Barry Roux, for Pistorius, said the law was correctly applied in convicting and sentencing.
Pistorius fired four shots through the locked door of a toilet at his Pretoria home on Valentine’s Day in 2013, apparently thinking Steenkamp was an intruder.
The State argued at the time that the court should reserve four issues for consideration, including whether the principles of dolus eventualis were correctly applied to the case.
During sentencing on September 11, Masipa said it was important to ascertain whether Pistorius intended to kill when he fired four shots at the locked toilet door.
Someone found guilty of murder dolus eventualis would have foreseen that their actions would cause someone’s death, but carried on regardless.
Nel argued that Steenkamp was shot dead in “the most horrendous circumstances”.
Pistorius was sentenced to a further three years, suspended for five years, for discharging a firearm at Tasha’s restaurant in Sandton, Johannesburg, in January 2013. The State did not include this in the intention to appeal.