Before he started racing Henri François Béconnais competed in cycling and motorbike races. In cycling he was a French champion in 1890 and, as a motorcyclist, was the outright winner of the Carrera en Cuesta de Chanteloup at the end of 1899, beating all the other 106 entrants, which included several different car classes, tricycles and motorcycles, with his Aster.
In the First Tour de France Automobile 1899, he was 6th in the three-wheel class with his Aster. He was also one of the first people to set speed records on a motorcycle. On March 30th 1900, at the Semaine de Nice (Nice Speed Week), he set a speed of 91.8 Kph with a Soncin.
Béconnais continued competing with motorbikes and cars. In June 27-29 1901 in the Paris-Berlin race, a distance of 1105 km, subsequently named VI Grand Prix de l'A.C.F. Beconnais failed to reach Berlin.
May 16, 1903 the VIII Grand Prix de l'ACF. Paris Bordeau Madrid. 275 cars participated. Beconnais' Darracq competed in the voiture légère class (light vehicle) but he crashed and was injured. The race was stopped at Bordeaux, after 550 km due to fatal accidents (Barrow died in hospital June 13th while his mechanic Pierre Rodez was killed instantly). As a result open road racing in France was subsequently banned. In June 1903 at the Circuit des Ardennes a six lap race totaling 512.05 km Beconnais' Darracq finished 6th.
In the French Gordon Bennett eliminations in May 1904 at the Circuit de l'Argonne, Béconnais retired on lap 4. Later that year, on the 2nd July, Béconnais was practicing for the upcoming Gordon-Bennett race on public roads near Labouheyre, to the North of Bayonne in southwestern France. On the Subida de La Bouhèyre a back tire burst on his Darracq 100 HP.
He lost control and hit a tree so hard that the engine ended up over 10 meters away. Béconnais perished as did his mechanic, Jules Bernard. In their memory, the newspaper "L'Équipe" erected a memorial to them at the scene of the crash.