A crossbreed between a Jack Russell and a French Poodle is known as a "Jack-A-Poo" or "Jackadoodle." The first is sometimes written as "Jackapoo" or "Jack A Poo." This type of dog is a hybrid and not a purebred dog.
According to DogBreedInfo.com, not all hybrids have two 100-percent purebred parents. The site also warns that dog owners who own a hybrid dog "can get any combination of any of the characteristics found" in either of the dog's parent breed. Prospective dog owners should research all the behaviors and temperaments associated with the dog's parent breed to determine the possible temperament of the hybrid.