A Brief History of the Marque
Charles Stoppa opened a motorcycle shop at Prado Place in Lyon in 1928 and began building motorcycles under the Stop brand in 1931 using JAP engines from Moto-Comptoir and locally made components sourced mostly from St Etienne. A limited number were produced.
It seems likely that the firm also produced sidecars as it is reported that Steib bought the sidecar firm "in the 1930s", and there is a 1933 German sales brochure for Stoppa sidecars. From this one could deduce that Steib marketed the Stoppa under its original name, probably until 1939.
Alexis Stoppa was the brother of Charles and opened a store named Alexis-Moto, later run by his daughter and grandson. 
Notes: 1. The Stoppa brand for the motorcycles is referred to by some sources. The correct name is "Stop".
2. alexismotos.com writes "In 1973, the company was taken over by Mr and Mrs Etard. In 1996, their two children, Pierre and Christine succeeded them. Like his grandfather, Pierre Etard, current CEO, shares the same taste for two wheels. He entered the Paris-Dakar race. His son Nicolas followed in the same footsteps by participating with him in two races. Today, Nicolas and Guillaume take over the family business."
Sources: François-Marie Dumas, alexismotos.com, et al.