Weber, who has been active as a Nordic combined athlete since 2003, began, like his role model and today's Oberwiesenthal training partner Eric Frenzel, at the Geyer Ski Sports Club. With Jens Einsiedel and Frank Erlbeck, Weber had the same coaches as Frenzel.
Weber made his international debut at the Nordic Ski Games of the OPA 2011 in Baiersbronn , where he narrowly missed the medal ranks both in the student individual and with the team. In the following years he competed regularly in the Alpine Cup until the end of 2015 . There he was able to draw attention to himself with good results and a total of six wins. With six podium finishes at the Alpine Cup in a row, Weber traveled to the 2014 Junior World Championships in Val di Fiemme . In the individual competitions he achieved the ranks 8 and 17, whereas he won silver with the relay around Dominik Schwaar , Jakob Lange and David Welde . A week later Weber made his debut in the Continental Cup in Klingenthal and was able to collect a few points straight away as thirteenth. In the summer of 2014 Weber was even allowed to try his hand at the highest level when he competed in the Grand Prix for the first time in Oberwiesenthal . In his third competition he managed to win his first three points. At the Junior World Championships 2015 in Almaty Weber again finished second with the relay.
At the beginning of December 2015 Weber made his debut in the World Cup in Lillehammer . Again he showed hardly any teething problems and immediately reached the point ranks with seventeenth place. Nevertheless, he was first downgraded to the Continental Cup , where he celebrated his first podium finish in Pyeongchang in mid-January 2016 in third place. Due to further strong results, he was nominated for the third time in a row for the Junior World Championships 2016 in Râşnov , Romania , where Weber won the bronze medal in the 10-kilometer Gundersen competition on February 23, 2016 behind Bernhard Flaschberger and Vinzenz Geiger . Two days later Weber followed up with silver in the sprint before he won silver again in the Gundersen team together with Martin Hahn , Tim Kopp and Vinzenz Geiger. After Weber presented himself in strong early form in the 2016 Grand Prix and, among other things, narrowly missed the podium in fourth in the Gundersen Singles in Villach , he also started at the highest level in the World Cup in winter . There he was almost exclusively part of the German World Cup team. He finished the season in 30th place overall. In the winter of 2017/18 , Weber faced the competition throughout the World Cup and was ultimately 31st in the overall ranking with 159 points.
At the German Championships 2018 in Hinterzarten , Weber was third in the individual and runner-up with Eric Frenzel in the team sprint. In the 2018/19 World Cup , Weber repeatedly achieved strong performances and, for example, reached the top 10 in Ramsau at the end of December 2018 in seventh place. Therefore, he was appointed to the German squad for the 2019 Nordic World Ski Championships in Seefeld , where he only won the Gundersen singles Normal hill was used. Weber came in 24th. In the overall World Cup ranking, he finally came in 23rd. His strengths were clearly in ski jumping, so that he finished the special ranking for the best jumper in the combination in tenth place, whereas in the trophy for the best runner it was only 36th place was enough. At the German Championships in 2019 in Klingenthal and Johanngeorgenstadt , he and Frenzel again took second place behind their Baden-Württemberg colleagues Manuel Faißt and Johannes Rydzek . The start of the 2019/20 World Cup season was a bit bumpy at first, although he already had a strong position with eleventh place in Lillehammer.
Terence has a sister, but is the only competitive athlete in his family. Weber is a sports soldier.