Hi Maread,
I was in a similar situation, so I know how awful it can be.
Have you called up to the other apartment, to let them know you are being disturbed, and to ask them to be quieter during the night? I would start off by doing this.
Then if the situation does not improve, get your landlord to do the same, and also to send them a letter outlining the problem and asking them to be quieter.
Are they renting or owner-occupier? If renting, you could contact their landlord aswell.
What kind of noise goes on all night - loud music, or dropping things on the floor etc?
Also, have you tried earplugs - they dont block out all noise, but do reduce the volume.
If the other tennant is up all night, (s)he'll need to sleep during the day, so I'm sure someone will suggest you turn your music on full blast before you go to work in the morning and leave it on all day!
Best of luck with it,
Ragazza.