Black spot and mould on rose

Hi Everyone, I'm new to this forum and would be grateful for some advice. We've recently moved into an old house with a huge neglected garden so we're learning about gardening as we go along! We have a large climbing rose which looks to be many years old and I gave it a hard prune in the Autumn when we moved in as it was very straggly. It now has a lot of new growth as well as the old ones but the vast majority of leaves are covered in black spot and a mould type bloom and a lot of the rose buds have the mould on them as well. It is flowering but the flowers don't seem to last long before they die.  

I have sprayed it thoroughly with a rose pest spray but it doesn't seem to have helped.  My question is whether, because the disease seems so entrenched, should we just get rid of it? Another rose close by also has the black spot and mould so I don't know whether it spreads.  

I don't like the thought of getting rid of an established plant but would this be the estate thing to do? Thanks