If there isn't too much try what my son did with some success: Wait until it has grown a bit and then cut off the individual stems and carefully pour deep root weed killer down them (they are hollow). This plant is incredibly deep rooted and digging it all out permanently is well nigh impossible. Burning the tops will have no effect at all except to stop it seeding!

Watch out for it appearing elsewhere in the garden - young plants in new places can be dug out if you get them before they are well established and get their roots down deep. Be merciless and vigilant and the best of luck!

Because of the problems associated with Japanese knotweed it is better to assume that other plants are going to be harmed by chemical weedkillers. You could dig out the desirable plants now and plant them in a new bed, making sure that you wash all the soil off the roots to get rid of any bits of knotweed. Glyphospate weedkillers are recommended (eg roundup) but I believe that there are others. You will probably have to treat over several years before you get rid of the knotweed and late summer is the recommended time to treat it (Environment Agency - http://www.environment-agency.gov.uk/homeandleisure/wildlife/130079.aspx#How_to_control_Japanese_knotweed)

I would say that Steragram's method is the way to go. Let the Knotweed grow to about 12 to 18 inches high. Cut the stem below a leaf node. This will give you a hollow stem. Take a pipette and load it with a concentrated solution of Glyphosate and squeeze it into the stem. Then wait to see the weed die down. You might have to repeat the process.

Thanks for the advice. I shall try the stem treatment and keep my fingers crossed!

I wouldnt wait till they are 12" tall though, go for 6-9". they will have had less chance to put on root growth. come the late summer treat any small bits you see as the plant will translocate the weedkiller down to it srhizomes and do a lot of damage over the winter. you may have to keep repeating this over and over though.

I think Bulbaholics suggestion of lifting the plants you want a good one. I did this to rid a border of ground elder. A pain having everything in pots for a year but worth while inthe end.