There was a hard and fast rule in the dugout of the University of Memphis baseball team: no cellphones. But on a Sunday in April 2006, coach Daron Schoenrock allowed his No. 2 starting pitcher to bring his cellphone to the game.

Schoenrock had told right-hander Stephen Gostkowski to take the day off. Gostkowski’s family members were in town, and Sunday was looking to be an eventful day for them. Gostkowski would have none of it. There was a doubleheader against visiting Mississippi Valley State and he wanted to be there, even if he was not going to pitch.

During the Tigers’ doubleheader sweep, Gostkowski’s cellphone rang and he went outside the dugout to take the call. He returned a few minutes later.

“He was as white as a ghost,” Schoenrock recalled this week in a phone interview. “He said, ‘I just talked to Bill Belichick. The Patriots took me in the fourth round.’”

They had indeed — at No. 118 overall — one of two kickers selected in the 2006 NFL draft. The other was Virginia’s Kurt Smith, who went to San Diego in the sixth round.

Smith’s next kick in the NFL will be his first, while Gostkowski is New England’s career leading scorer.

Gostkowski was originally a walk-on for football at Memphis — he was given a baseball scholarship — but won the kicking job as a freshman.

“He was a very average kicker in high school with a strong leg who didn’t always know where the ball was going,” said Tommy West, his football coach at Memphis. “But you still saw something there. And he was a determined kid.”

Over four years, Gostkowski made 70 of 92 field-goal attempts and 159 of 165 extra points. He showed his versatility by recovering his own onside kick and his foresight by using an NFL-size tee on kickoffs. He left Memphis as the 13th most productive scorer in NCAA Division I-A history.

Schoenrock thought football was Gostkowski’s best bet for the future. Nonetheless, baseball scouts had inquired about Gostkowski, impressed by his ability to throw hard sliders in the 90-mph range and apparently undeterred by his career record of 7-22 with a 6.04 ERA.

“It was harder for me with baseball,” Gostkowski said. “I didn’t do it year round.” Because of football, he missed the fall workouts with the baseball team. In the spring, during the baseball season, Schoenrock had his pitching coach bring a bag of footballs to practice.

The Patriots were among a handful of teams that worked him out at the Liberty Bowl. They were about to part ways with Adam Vinatieri, their kicker since 1996.

“I had no idea what was happening,” Gostkowski said. “They don’t give you any inclination. I thought I did pretty well. I didn’t do the combine. I went to the Senior Bowl. I pitched my last game a week before the draft. I was in my own world.”

He is still very much in his world, one in which he has surpassed Vinatieri, a likely Hall of Famer, as the Patriots’ scoring leader. His accuracy is astounding. He has converted an NFL record 439 consecutive extra points since his only miss, during his rookie season. He has made more than 87 percent of his field-goal attempts, including all 17 this season. Among them is a career-best 57-yarder.

Not only is he the leading scorer in franchise history, he is No. 2 in continuous service behind Tom Brady and, at 31, second in age to Brady. A case could be made that not only are they the two oldest players on the Patriots, they are also the two most secure. Belichick has not even bothered to bring in a kicker during training camp to push Gostkowski.

“One of the things New England asked him when they worked him out was whether he could take tough coaching,” West said. “You know, sometimes you don’t want to be too hard on kickers because they can go into a shell. With Stephen, the more you got in his face, the madder he got. And the madder he got, the better he kicked.”

Gostkowski said: “I’m proud that I’ve been able to stick around in an environment where the media is tough, the fans are tough, the weather is tough and the coaches are tough. When you’re on a good team, you just want to do your part and know that they trust you.”

That trust is implicit now. But in Super Bowl XLII against the Giants, during Gostkowski’s second season, Belichick chose to go for a first down on fourth-and-13 at the Giants’ 31-yard line midway through the third quarter. The Patriots lost the ball on downs. The Giants won 17-14.

Since then Gostkowski has become as omnipresent as a kicker can be. He has led the NFL in scoring four times, including the last three seasons.

“As kickers, we don’t make our own opportunities,” Gostkowski said. “We take advantage of the ones we get. I don’t try to overthink it. I try to go out and make every kick I can and if I miss, I’ll try to make the next one.”