Peugeot UO-9 super sport rim upgrade
What you have there is a freewheel, not a cassette. The difference is a freewheel combines the freewheel ratchet mechanism and sprockets and mounts on a large diameter thread on the hub; the freehub/cassette system puts the freewheel ratchet mechanism in the hub and the sprockets slide on a splined freehub body. See here for in depth explaination.

You probably have a hub with frame spacing of 120mm (see here). Modern 700c road wheel have a spacing of 130 or 135mm.

What you will want to do is get a wheel with a 120mm freewheel type hub onto which you can put your existing freewheel (or a new one), but with a 622mm diameter rim (700c) rather than the 630mm (commonly called '27 inch') you have now.

I assume you know you will have to find long-reach rim brake calipers to deal with the difference between rim diameters.