The structure and properties of a geometrically constrained magnetic wall in a constriction separating two wider regions are studied theoretically. They are shown to differ considerably from those of an unconstrained wall, so that the geometrically constrained magnetic wall truly constitutes a new kind of magnetic wall, besides the well known Bloch and Néel walls. In particular, the width of a constrained wall can become very small if the characteristic length of the constriction is small, as is actually the case in an atomic point contact. This provides a simple, natural explanation for the large magnetoresistance observed in ferromagnetic atomic point contacts.