Leading 14-10 at halftime thanks to Liam Williams' opportunistic try and three Leigh Halfpenny penalties, Wales produced a strong defensive effort in the second half to restrict the French to just three more points despite their dominance of possession.
"I don't think he will go on tour in the summer", Gatland said. "It was an ugly performance in terms of it wasn't pretty, but we spoke beforehand that we just came here to do a job and the job was just to win and finish second in this competition".
It's going to be a close encounter and a very physical game.
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"We will start thinking about it in the next six weeks about which players we expect to take".
Wales failed to score a point after the 30th minute, leaving head coach Warren Gatland with plenty to contemplate ahead of summer appointments with South Africa and Argentina.
Scott Williams kicked the ball through, Trinh-Duc could not control it and Liam Williams dived over to put his team in front.
Martin, Baldwin lead Butler over Arkansas in Round 1
But believe it or not, the defense won this game for Butler, holding Arkansas' two all-conference guards for a combined 27 points. Butler has won in the first round of the NCAA tournament for seven consecutive times that they have made the tournament.
Another mistake by Trinh-Duc, this time a forward pass under no pressure whatsoever, handed Wales a ideal attacking position, but their scrum creaked again and Marco Tauleigne charged back up the field.
The visitors hit back when Fickou, who started a nice collective move, ended it with a try after Machenaud, who converted, and Trinh-Duc played quickly to send him over.
The French scrum-half, though, missed a penalty on the stroke of half-time, and Wales held a four-point advantage at the midway point in the battle for second place.
Wales, despite the odd charge from outstanding number eight Taulupe Faletau, could not establish any kind of territorial foothold in the French half, and even with 20 minutes to go, it looked as though they were hanging on.
Sweden says happy to help on North Korea, amid summit speculation
There has been no official word from North Korea since the invitation was delivered via South Korean officials in Washington. A source familiar with the matter declined to comment but told Reuters the bilateral talks would end on Friday.
Wales were penalised for coming in at the side of the resulting ruck, but Trinh-Duc - with Machenaud having been substituted - missed a sitter of a penalty to sum up his less than average afternoon on the paddock.