If the plant is dying (of natural causes, GRIN) the melons should be ripe. You have already indicated your sound test result. Another indication would be the condition of melon's stem. Is it semi brown, wrinkling? versu healthy thick green???
At any rate, leaving the melonss on the vine longer is not going to hurt, unless there cretures out there that like them more than you do.

I have the same problem - the plant appears to die (leaves turning pale green with some brown/rust spots all over them) but the stem of the fruit is still fully green and not dry at all. We had lots and lot of wet and cloudy weather around Richmond, so I am not sure what to think.
couple weeks ago:
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now:
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The stems are still green.

Jollyrd, my plant appears very similar to yours, though is even worse. No weather fluctuations here, just hot and sunny, and I actually cut back on watering.

The stems shou;d be vibrant and green. The indicator is the tendril where the stem attaches to the vine. It should be completely dry. Sugar Babies are tricky, so use all the indicators for ripeness, Yellow where it lays on the ground, rough skin with no filmy coating, and the thump test ( make sure you have a comparison like a known unripe melon to thump)

The stems on our Sugar Babies stayed green. I checked and they were yellow on the bottom side, and I had seen in another thread that someone said you need to watch timing on Sugar Babies. Ours had stopped getting larger and had been growing a couple weeks longer than the estimated maturity date. Decided to try them and they were ripe and very good. But the stems were very green still. (Jollyrd, ours vines also looked like yours. I thought it was some sort of blight, happened fairly late in the maturity of the vines, but didn't seem to affect the watermelons. Vines stayed healthy, just the leaves appeared to be affected.)

Funny thing - I just logged on to GardenWeb to ask something similar. I have (had?) six melon vines growing well with lots of fruit almost ripe and out of the blue a foliage disease (I believe) spread through them and within 7 days all the leaves where wilted, brown and gone.

To my inexperienced amazement the fruit continued to ripen and I have harvested a Noir de Carmes, Collective Farm Woman, Honeyrock and Petit de Rennes. They were of average quality, but passable.

I really thought without leaves they couldn't ripen. Maybe one difference is that the leaves are dead but not the vine. The vine still looks normal as far as I can tell.

well, I went to the garden this morning, - before I read the reply posts - and guess what? One of the babies was cracked and I can see the red and I can smell the goodness sweet aroma! Had no time to taste it, had to run out to work. Will take pictures and tell you how it tastes tomorrow.
If this one tastes good, I might go ahead and eat the other one that is similar size

the first watermelon (that was cracked in the garden) was almost 13lbs, 8 in diameter, taste - so-so - sweet enough, very watery.
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Then I got the next one - also 12lbs - it was not cracked. I chilled it before cutting and eating. VERY good taste, not too sweet.

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Also cut my one and only cantaloupe - over 2 lb, - it was sweet but not very flavorfull.

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Having the same issues. I'm in Ga. Very hot here. My leaves and vines look good. I planted 4 seeds and so far I have 4 good looking sugar baby watermelons. However, after the first 4 came out, all the rest of the tiny ones started dying. From the time that I planted the seeds it says my harvest time should be August 5th. OAN: I put the potentially good watermelons on soil and paper plates because I thought maybe the sun was burning them on my deck and killing them of. Please help :(

More pics

More pics

i have water melons growing next to cantelope all of a sudden my melons are wilting and dieng with only 2 melons are the being killed by the catelope they are withering up overnight

I have 2 types of melons growing in my garden. Sugar Babies and another I've forgotten what they were (lost tag) but my sugar babies have only grown just over the size of a softball and vine has browned and dried out, still connected to the fruit but haven't gotten any bigger. Are these considered trash or what can I do to help them or have better results next year? The other melons in the garden have also seemed to stop growing, although a few smaller ones have started to grow

Picture didn't post but here they are