You have read about these before when I have mentioned about them.
Most people who replace belts on their machinery, usually put a new v-belt on them or if they have a difficult installation, maybe a link belt. I have been using this product in industry for at least the last 20 years, and in my old shop I used this type of belt on both my mill and lathe, and found it superior to everything that I had used before.
I don't know if it is available in other parts of the world under a different name, but here in the UK it is called Redthane (or as I have now noticed, Greenthane). I buy it from my local engineering supplier, who always has it in stock (that shows just how popular it is with the maintenance guys).
http://www.poly-products.co.uk/beltext.htmI purchase it in multiples of metre lengths, and it comes in a variety of sizes. The normal ones I use are 10mm or 12mm. It costs about 7 squid a metre, roughly the same cost as a good quality v-belt. I think it goes down in size to about 5 or 6mm for those really small pulley drives, and of course a lot larger, for real big stuff. The main advantage is that it can easily be made into any belt length that is required and can be joined AROUND say a lathe spindle rather than having to do a strip down to change the belt.
This shows what is left of my little stash, with enough left to finish off the die filer and the very long belt that is used on my surface grinder.

The die filer really requires a 12mm belt size, but because I don't have enough left, 10mm will be perfectly OK. As long as the belt doesn't touch the bottom of the groove, it will work perfectly well.
What I do is pull the belting around the pulleys as tight as I can, if you notice at the back of my hand, around the small pulley it doesn't look too tight, this was as tight as I could pull it. I marked it up with a gap of about 1.25" (32mm). The belt was cut at that mark.

This is a major component that is required to weld the two ends together. This is just a piece of steel plate with a twist in it to put it into the correct position for use. Also required is a blowtorch to heat the top end of the plate up to a very dull red colour.

Because I don't have three hands, it is very difficult to show you how the next bit is done, so I will have to explain it to you.
A bit of imagination has to be used here.
First, the plate is brought up to dull red, then the torch is taken away and using TWO hands, the belt is allowed to form into its natural round shape and once in that position each end of the belt is pushed against each side of the hot plate. It needs to be held there for about 5 seconds, during which time the ends will melt and start to 'run', you can see previous runs on the plate. Then in one motion, still with the ends being pushed towards the plate, both ends are slid up the plate and off the end, and the two melted ends are pushed together.

I then transfer it into a bit of old v-groove stock, you could just use a bit of angle iron. Then keeping pressure to push the joint together and also down into the v groove. I am showing it being done with two fingers, you would do it with two hands, it isn't because I am superman, just that I am using my other hand to take the piccy. Hold it in that position for about five minutes (if you can).
By using two pairs of hands, it is just as easy to assemble the belt around a shaft, as I mentioned before, saves having to strip down the lathe spindle. The spare pair are there to warm up the plate and hold it while you push the ends against it, and get things out of the way when you are finished with them.

Do NOT pull, push, stretch, twist, poke, or even fart near the joint at this stage, just place it gently onto the bench for 15 to 20 minutes for the joint to cure.

After you have waited the allotted time, then you can pull, push, stretch, twist, poke, or fart on the joint. If it doesn't come apart, you have got a good weld, if it does fall apart, then repeat the exercise. I have about a 99% success rate now. When I first started doing it, I got a few more failures than I really wanted. It is all to do with the melt, slide and hold bit.
Assuming all is well, carefully dress away the bulgy out bits with a very sharp knife or razor blade. Job done.

So that is two 12mm ones for my lathe and a 10mm one for the die filer.

Now fitted.
But it has couple of advantages that normal v-belts don't possess.
The first being that to change to a different pulley size, just stretch it over to the new pulley, no need to slacken off the motor and retension.
The next is that I have never had one of these belts slip, even when covered in grease and oil, as long as it has tension on it, it will drive, and unlike what you maybe think, it doesn't act like a big rubber band, and 'twang' all over the place, in fact I find that it smoothes out machine drives and makes them quieter.

I was going to make and fit a tensioner onto my lathe to allow me to swap over the belts between the speed ranges, but now, I will just stretch and reposition.

These belts can be bought as two types, either solid all the way thru, or with a hole in the middle, as in tube. If you are welding it, you can use either one, the solid would give a much stronger joint (I use the tube, and have never had a joint fail in use in my shop, I can't tell you if any other of my joints have failed, as I no longer work for the companies). But if ever a joint did fail, just reweld it back together again.
For the tubular type, you can buy special little barbed couplings, that joins the two ends together. I have had those joints fail after a time in industry, so I would always recommend the welding technique if possible.
Bogs