Experts and BHG readers answer.
The usual morning glory (Convolvulus) is an annual vine. It grows, bloom, and dies all in one year. With luck, it will self sow and new plants will come up from the seeds from last year's vine. All you need to do is pull up the old vines and dispose of them.
If yours is growing into another year, you probably have one of the Ipomoeas, perennial or evergreen morning glory. It grows from a tuber, like a sweet potato (which it is related to) and from long runners that will root to form new plants. It is very vigorous. In some areas of the south it is considered a noxious weed. Some even compare it to kudzu, the "vine that covered the south". It can be controlled by restricting water and bold pruning. Now is the time to clear away all the old vines that died over the winter. The plant will come back from the tuber.