Which structure or organ is thought to be responsible for stopping the eating response?

(Ventromedial) hypothalamus

Among its other functions, the hypothalamus controls a person's feeling of hunger and satiety in two anatomically distinct centers -- the lateral hypothalamus and the ventromedial hypothalamus. When glucose levels in the body are low, the liver signals to the lateral hypothalamus that a person needs to eat to increase his/her glucose levels. This gives the feeling of hunger.

As a person eats, he reaches a point called satiety, which tells him/her that he/she should stop eating. As food begins to move from the stomach to the intestines, a hormone called cholecystokinin (CKK) is produced to signal the ventromedial hypothalamus that a person must stop eating. This gives the feeling of satiety, often characterized by the full or distended feeling of the stomach and intestines.