Nice boost for Rhein at home of golf
Gibson shot a two-under 286 over the four rounds but it wasn't enough to defeat Scotland's Keir McNicoll, who finished at five under after hitting 283.
Troon golfer Michael Stewart was equal third (284) with Frenchman Rudy Thiullier, while Sam Hutsby finished fourth with 285.
It was Gibson's first foray on the course that is dubbed golf's spiritual home.
The Links Trophy has been contested since 1989 and boasts former winners Justin Rose, Richard Finch and Oliver Fisher, who now all strut their stuff on the European Tour.
Australian Stuart Bouvier is the only player from outside the United Kingdom to have won the event.
Gibson has just completed a scholarship with Oklahoma Christian University where he gained All American honours four years running.
The British Amateur Championship will be contested at Turnberry from June 16 to 21.
The first two rounds are contested by strokeplay with the final rounds decided by matchplay.