No. The era of the Judges ended precisely when Saul was made king by the last of the Judges (Samuel). See also:The JudgesThe KingsSamuel
Abraham then king David then MosesAnother Answer:King Saul ruled before King David and King Solomon ruled afterwards. All three ruled for a 40 year period each.
David became king after Saul's death around 1010 BC. Following Saul's demise, David was anointed king over the tribe of Judah, and later, he became the king of all Israel, uniting the tribes under his rule. His reign is marked in the biblical narrative in the books of Samuel.
David was King in Hebron for several years until the death of Saul. During that time, Saul pursued him intermittently, believing that David threatened his (Saul's) monarchy. At one point, David had to hole up in the territory of the Philistine King Achish. When Achish's men wanted to kill David, he pretended to be insane and they let him be.
Ish Bosheth, the son of Saul, ruled the northern tribes for two years after Saul, while David at the same time ruled Judah and later became king over all the Israelites.
David was anointed king while Saul was still in charge because God had rejected Saul as king due to his disobedience and failure to follow divine commands. The prophet Samuel was instructed by God to anoint David as Saul's successor to prepare for a transition in leadership. This anointing was meant to signify God's choice of David as the future king of Israel, despite Saul remaining in power at that time. The situation ultimately set the stage for the conflicts and events that followed in the biblical narrative.
The people had asked Samuel to appoint a king (1 Samuel ch.8). At that time, Saul's family had lost two donkeys, and Saul went out in search of them (1 Samuel ch.9). God informed Samuel through prophecy, that a man would come to him the next day, and he was to be made the first Israelite king. The following day, Saul's search brought him to Samuel's town, and Samuel anointed him as king. See also:More about SamuelAppointing the first kingSaul's accomplishments
The Ark of the Covenant was captured by the Philistines from the Israelites (until they returned it several months later) and was displayed by them for a couple of days in their Temple of Dagon. This had nothing to do with King Saul - it was before his time.
Sauls time was not so good, but king Davids and king Solomons period were the best in Jewish history.
King Saul and Jesus were not contemporaneous: Jesus died about 32 or 33AD but King Saul died about 1010BC. However, Paul the Apostle came from Tarsus and changed his name to 'Saul 'upon his conversion.
The prophet at the time of king Sauls reign was the major prophet Samuel he was also there at the time of king David.
It was God who commanded that David be anointed at that time (1 Samuel ch.16), since Saul had been lacking in fulfilling one of God's commands (1 Samuel ch.15). Saul had spared the Amalekite King Agag in battle, against God's instructions through the prophet Samuel (ibid). See also:More about King DavidMore about Samuel