Looks like fuel return line to me. So the float in the carb must be set high and unused or excess fuel in the bowl is sent back.

Dodgey way is to block it or cut a t peice into the fuel hose to create a loop. The factory dual line rail does this too, but you would be replicating same thing.

I took the hose off and took a photo

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What sort of weber?
If its a downdraft DG#V it may have a loose/leaking pressed in fitting into the alloy body. If so then get it removed and the hole tapped for a threaded sealing plug.

These suckers can and do fall out and spray raw fuel on your exhaust manifold. Not good.

BTW - Webbers are in Red Bull F1 cockpits. Webers mix fuel and air.

If you are using a regular fuel pump for a Weber, it is only 3.5 psi -- not high pressure. No return line is needed.

Pics of the hose. Done carbs have an.overflow Return line. Need to work out what you have.

Could be tank.pressurising in. The heat, make sure your cap still vents at the fuel filler too.

Looking in the engine bay the other day, I realised that the webber was leaking fuel out of a blocked off hose. I thought the hose was worn so put a new piece of hose on there but it didn't fix it.
I suspect that there is a large amount of pressuring building up and forcing the fuel out

Anyone have any ideas? Should there be a blocked off hose on the webber? =/