Complaints of ankle weakness or pain with running is a fairly common problem for many runners. It usually occurs in those who have a history of spraining their ankles, in which case it occurs because the ligaments of the ankles remain somewhat damaged after the sprain. However, it can also occur in the absence of any history of an ankle sprain.
There are several possible solutions that you could explore. One would be simply to switch to a lower impact form of exercise, such as swimming or elliptical training, as these will be much less likely to cause discomfort in your ankle.
Another solution would be to make sure you are wearing the right athletic shoes for your body type. If your feet have a tendency to pronate or if you need additional arch support, then wearing an appropriately designed shoe should help with this. Many athletic shoe stores can help you analyze your running gait and determine which is the best shoe for you.
Finally, you could look up ankle stabilizing exercises or ask your personal trainer about them. These are a series of simple exercises designed to build up strength around the joint and prevent injury
If you have significant pain, swelling, or other serious symptoms, talk immediately to your primary care doctor