Yes - cut the finished stem back to just above the node nearest to the base. The new stem might grow from there, or else from between the leaves. Obviously, treat the other stem the same once the last flower has fallen off.

Some people leave the whole stem on, and hope that a new one will emerge higher up - I've found that all that happens is that the stem looks ugly and goes brown and brittle - so it gets cut off eventually!

Thats what i call a qick reply spritzhenry,Ihave been looking at it for acouple of weeks now,not daring to risk it as it was a gift and it is really beautiful,while i am online could you advise me on repotting and feeding, thanks very much

If it's a Phalaenopsis orchid, I cut just above the first node below the lowest flower, and I usually get one or two new stems from there. Most other kinds of orchids aren't so cooperative, and wait to bloom next year.

Blencowe, you shouldn't need to repot it for at least two years, and when you do, use special orchid 'compost' which is like bark. Feeding - I have a little pot of power which I put in the water, and water on once a month, not more. I got both of those at a Garden Centre.

My rules of thumb:
Don't repot until there are more roots outside the pot than in.
Use a soluble, high nitrogen, houseplant fertilizer at 1/3 the recommended strength, about every third watering.
Hope that helps!

thanks Tugbrethil there are a couple of roots hanging out but i would sey there are more in the pot,but it does seem a very small pot for the orchid,is this usually the case.

Yes - they're always sold in small pots...not a problem. I have mine in their see-through plastic pots placed inside a ceramic pot to stop them toppling over as well as aesthetic reasons!

Some people say that the roots should be exposed to the light, not put into outer pots, but I haven't found it to be a problem at all - I also checked with a good Orchid Nursery - and they said that was fine.