The problem will still happen even with a new clematis. You could reduce some of the old brown stems down to a few inches from the floor then you would get new growth from the bottom. and as you have only done a few stems the rest would still flower higher up.

I'm afraid Seaburngirl is right. You could always try underplanting with something like Virginia Stocks, which might hide some of the brown stems.

Having agreed with the above, I would also suggest that you try a vigorous Herbaceous type Clematis which you can cut right down to the ground and they come back fine each year.That said, the flowers are smaller than the type 3 Clematis, but there are some beautiful varieties

Thank you, everyone. It obviously depends how high up the browning tends to go. I have distant memories of seeing Ville de Lyon brown pretty much four feet up from the ground, that would make it difficult to conceal with shorter plants. Maybe indeed some sort of herbaceous clematis would work--I'll look into that and let you know how I get on!

I have a lovely blue flowered herbaceaous one called Alan Bloom. Its in my plant profile with pictures.