From the summary on wikipedia: " Ryuk realizes Light has lost and uses his personal Death Note to kill Light with a heart attack, as he had promised to do one day at the beginning of the story." <--- So I did not remember totally incorrectly :)
And they do both die in the movie, and once again Ryuk is the one to kill Light: "It is revealed that L is alive and wrote his name in the Death Note with the intent to die almost a month in advance, thereby foiling Light. The task force did not leave for the U.S. but instead surveilled Light from another location, therefore hearing and observing Light's confession as Kira and replacing Light's Death Note with a decoy. Cornered, Light pleads with Ryuk to kill L and the task force. Instead, Ryuk writes Light's name in the Death Note, telling Light that anyone who uses the Death Note is banned from heaven and hell and will instead spend eternity as nothingness. Light dies in his father's arms, begging him to believe he acted as Kira to carry out justice.
Knowing that supernatural events were involved, the official line put out by L and the investigation team was that Light was killed by Kira. Twenty days later, Soichiro and L meet one last time before L dies peacefully. One year on (on Light's birthday) Soichiro keeps to the lie to Sayu and his wife, as Sayu adores Light but did not support Kira's actions. Misa also celebrates Light's birthday, still loving him but has no recollection of the Death Notes."
no subject
And they do both die in the movie, and once again Ryuk is the one to kill Light: "It is revealed that L is alive and wrote his name in the Death Note with the intent to die almost a month in advance, thereby foiling Light. The task force did not leave for the U.S. but instead surveilled Light from another location, therefore hearing and observing Light's confession as Kira and replacing Light's Death Note with a decoy. Cornered, Light pleads with Ryuk to kill L and the task force. Instead, Ryuk writes Light's name in the Death Note, telling Light that anyone who uses the Death Note is banned from heaven and hell and will instead spend eternity as nothingness. Light dies in his father's arms, begging him to believe he acted as Kira to carry out justice.
Knowing that supernatural events were involved, the official line put out by L and the investigation team was that Light was killed by Kira. Twenty days later, Soichiro and L meet one last time before L dies peacefully. One year on (on Light's birthday) Soichiro keeps to the lie to Sayu and his wife, as Sayu adores Light but did not support Kira's actions. Misa also celebrates Light's birthday, still loving him but has no recollection of the Death Notes."