Depends what you mean by fast-growing! I would choose Yew as you can cut it back if it gets too large, unlike most conifers.

Thanks Landgirl . . . I will have a think!

Hi Sheila! You've just described my Sparten Junipers to a tee. Have a look and see if they interest you. I love these because they grow in a narrow columnar (I can still park my car in the drive). No pruning and they move gracefully in the wind. They just want full sun and soil that drains. They are very fast growing - topping out at about 20-25 feet.

http://www.growsonyou.com/photo/slideshow/332616-spartan-junipers/member/bathgate

Many thanks Bathgate, but I'm afraid this spot is in full shade. I'm actually wondering about planting Bamboo!

Most prefer sun, Sheilabub, but Fargesia varieties tolerate a fair bit of shade, and, luckily, are clump forming, not runners. The clump expands over time though...

Cotoneaster Conubia is a semi-evergreen which would look nice but if you want something that will go big and fast then Thuja Plicata will fit the bill. Having said, I am a great fan of Landgirl's suggestion of Yew but will be expensive to buy in a large specimen. However, you do have to consider that a fast-growing tree or shrub my end up being an unmanageable beast.

Thank you Bamboo and Jimmy. We have a good man working here now, and he is suggesting variegated Leylandii because there is already some at the back of the garden. Away from the border, I'm thinking of planting a mature Betula jacquemontii as it will look good and the crown will partly screen the windows of the house behind. Just have to talk if over with OH!