Morris made his 11th first-class 100 - one ahead of Wayne Larkins - and hit Andrew Davies for five successive fours in racing from 100 to 150 in 39 balls. However, there was never any prospect of Durham setting Glamorgan a target, especially in the absence of their leading bowler, Melvyn Betts, who tore a groin muscle on Saturday.
Durham began the day 38 runs ahead and closed on 385 for 7 with David Boon on 62 not out. Morris hit 21 fours and four sixes in his 214-ball innings, ended on 163 when he was stumped by Ismail Dawood.
Glamorgan did not resort to declaration bowling, but they lost heart after Darren Thomas bowled Jon Lewis in the day's third over. The ball shot along the ground to hit the base of off stump, but it was one of few deliveries to misbehave in a match which, at that stage, had produced 1,122 runs for the loss of 22 wickets. Morris reached his 100 off 159 balls with 12 fours and two sixes.
The Durham vice-captain was also forced to issue a formal apology after finding himself under suspicion of making a two-fingered gesture to club members. But he explained: "It was directed at a mate of mine who bet me pounds 20 I wouldn't make a century. If the members feel it was directed at them I apologise."