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Saul committed suicide on Mount Gilboa.
1 Samuel 31:1 Now the Philistines fought against Israel; and the men of Israel fled from before the Philistines, and fell slain on Mount Gilboa.2 Then the Philistines followed hard after Saul and his sons. And the Philistines killed Jonathan, Abinadab, and Malchishua, Saul’s sons.3 The battle became fierce against Saul. The archers hit him, and he was severely wounded by the archers.4 Then Saul said to his armorbearer, "Draw your sword, and thrust me through with it, lest these uncircumcised men come and thrust me through and abuse me." But his armorbearer would not, for he was greatly afraid. Therefore Saul took a sword and fell on it.5 And when his armorbearer saw that Saul was dead, he also fell on his sword, and died with him.6 So Saul, his three sons, his armorbearer, and all his men died together that same day.
What happened after to his body after his death is recorded next.
1 Samuel 31:8 So it happened the next day, when the Philistines came to strip the slain, that they found Saul and his three sons fallen on Mount Gilboa.9 And they cut off his head and stripped off his armor, and sent word throughout the land of the Philistines, to proclaim it in the temple of their idols and among the people.10 Then they put his armor in the temple of the Ashtoreths, and they fastened his body to the wall of Beth Shan.11 Now when the inhabitants of Jabesh Gilead heard what the Philistines had done to Saul,12 all the valiant men arose and traveled all night, and took the body of Saul and the bodies of his sons from the wall of Beth Shan; and they came to Jabesh and burned them there.13 Then they took their bones and buried them under the tamarisk tree at Jabesh, and fasted seven days.
The men of Jabesh Gilead had special regard for Saul because he had previously saved them from the Ammonites (1 Samuel 11:1-11).Eventually David reburied his bones in the tomb of his father Kish.
2 Samuel 21:12 Then David went and took the bones of Saul, and the bones of Jonathan his son, from the men of Jabesh Gilead who had stolen them from the street of Beth Shan, where the Philistines had hung them up, after the Philistines had struck down Saul in Gilboa.13 So he brought up the bones of Saul and the bones of Jonathan his son from there; and they gathered the bones of those who had been hanged.14 They buried the bones of Saul and Jonathan his son in the country of Benjamin in Zelah, in the tomb of Kish his father...
The account that the Amalekite gives in 2 Samuel 1:5-10 is one of lies intended to curry favour with David. We can reasonably suppose from what he says that he had witnessed the conversation between Saul and his armour bearer. After they were dead he had taken Saul’s crown and bracelet and concocted his story to gain David’s favour and possibly a substantial reward.
2 Samuel 1:5 So David said to the young man who told him, "How do you know that Saul and Jonathan his son are dead?"6 Then the young man who told him said, "As I happened by chance to be on Mount Gilboa, there was Saul, leaning on his spear; and indeed the chariots and horsemen followed hard after him.7 "Now when he looked behind him, he saw me and called to me. And I answered, ‘Here I am.’8 "And he said to me, ‘Who are you?’ So I answered him, ‘I am an Amalekite.’9 "He said to me again, ‘Please stand over me and kill me, for anguish has come upon me, but my life still remains in me.’10 "So I stood over him and killed him, because I was sure that he could not live after he had fallen. And I took the crown that was on his head and the bracelet that was on his arm, and have brought them here to my lord."
He had misjudged David, who himself had refused to kill Saul because he was God’s anointed king, and paid with his life.
2Samuel 1:14 So David said to him, "How was it you were not afraid to put forth your hand to destroy the LORD’S anointed?"15 Then David called one of the young men and said, "Go near, and execute him!" And he struck him so that he died.16 So David said to him, "Your blood is on your own head, for your own mouth has testified against you, saying, ‘I have killed the LORD’S anointed.’"
The Amalekite died, not because he had killed Saul, but because he pretended that he had.The account of Saul’s death is given in the narrative of 1 Samuel 31:1 and 1 Chronicles 1-6 is the true one and not the story told by the Amalekite.
I hope this helps.