select your first country eg uk then set the from weight at 0 then the to weight at 9999 ( or make it smaller and leave all your product weights at 0.00 )
then set the fixed price you want
do the same for any other countries you are shipping to
v1.5 - remember to set the value of ShippingByWeightAndCountry.CalculatePerWeightUnit to false in admin configuration -- all settings (page 8)
Method described above that uses "Shipping By Country & Order Weight" module is actually quite complicated as it doesn't allow you to set 'default' rate for all the other countries. There is actually a simpler way to do this using By Order Total Module.
Say you have 3 rates US, UK and rest of the world.
Create 3 shipping methods let's call them "US shipping", "UK shipping" and "rest of the worlds shipping" for simplicity. Now on country restrictions list restrict everything except US for "US shipping", UK for "UK shipping" etc. Use the Shipping By Order Total module to setup shipping for each of the above shipping methods.
Viola! Try checking out something from your store with shipping addresses in different countries to test.