Your plant has no dry ends, which is an awesome sign. Watering a pony tail palm once a week, sounds like a lot. Still, you could have been correct to water that often if the soil is very fast draining. What soil is it in. If you did not add perlite or pumice to the soil, once a week watering would probably be more than the plant wants. In winter, I water mine about every two months. What is the pot made of. Terra cotta or something that allows air in as terra cotta does will dry faster, ceramic or plastic will take quite a bit longer to dry out. If your soil was dry almost all the way down to the bottom of the pot, before you watered then your watering was good. If not, I believe you have watered too much.
Feel all around the trunk of the plant for any area that is not as firm as it should be. These plants rot easily with too much water, and they rot within the trunk, slowly, so it is difficult to know what is happening. If you feel a slight spot that is not complete mush, I believe you may be able to stop it from progressing. I would hope for the best, in your shoes; but I would be very certain that the soil is dry down to 50 or 75 percent of the soil mass. These plants are easiest in very very bright sun, direct sun during summer may be a bit much. I would acclimate it to that much light if I wanted to try to grow it that warm. If your plant deteriorates more, please post again. There are other things you can do tho they are extreme. It does look good so far, so I would be optimistic.
