The late monarch's half brother Moqren was named crown prince, according to the statement.
King Abdullah, believed to be around 90 years old, was hospitalised in December suffering from pneumonia and had been breathing with the aid of a tube.
He died on Friday "at 1:00 am (2200 GMT)" and would be buried later in the day following afternoon prayers, said the statement.
In recent years, his advanced age and poor health had raised concerns about the future leadership of one of the world's key oil producers.
Abdullah's half-brother Salman, 79, was named crown prince in June 2012 following the death of Prince Nayef bin Abdulaziz.
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Salman had been representing the king at most recent public events because of the monarch's poor health.
In March 2014, King Abdullah named his half-brother Prince Moqren as a second crown prince, in an unprecedented move aimed at smoothing succession hurdles.
Moqren, who was born in 1945, is the youngest of Abdulaziz's sons.
Since the death in 1952 of King Abdulaziz al-Saud, the founder of Saudi Arabia, the throne has systematically passed from one of his sons to another.
But many of them are old or have died. The former crown princes Sultan and Nayef died in 2011 and 2012 respectively.