The nature of momentum?

I'm not a physics major or anything so I don't know much beyond the basic newtonian stuff taught in introductory college physics. One thing that's always bothered me though is how during collisions, momentum is always conserved but not kinetic energy. Wikipedia gives me this: "It can be shown that a perfectly inelastic collision is one in which the maximum amount of kinetic energy is converted into other forms. For instance, if both objects stick together after the collision and move with a final common velocity, one can always find a reference frame in which the objects are brought to rest by the collision and 100% of the kinetic energy is converted" My question is: if you start out with said reference frame, then won't the final momentum of the system be 0 also since the objects are stationary? What's wrong with my logic?