Answer: Based on your description I'm not sure exactly what happened, but there are two good possibilities. One is that a borer has entered the stem. If so, you will find "sawdust" nearby. You may still be able to pick out the borer by hand and then bury the damaged part of the stem so that it roots at the damaged site; this way the vine can continue to grow if you have caught it in time.
The other possibility is that cucumber beetles have destroyed the vine and it is literally rotting, possibly due to bacterial wilt. (Bacterial wilt causes a characteristic stringy and sticky sap in the stems -- you can squeeze it out and see it.) If this is the case, there isn't a lot you can do except try to control the cucmber beetles and hope to prevent infestation next year. Sevin is effective against cucumber beetles, especially if used early in the season; another important thing to do is to clean up, remove and destroy all of the vines and plant debris each fall to help prevent an infestation next year.
Good luck with your vine!
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