Ok, I found the problem. Apparnetly neither browser likes relative positioning, so what I ended up doing is changing
Basically I just had this block take up some extra space as filler, so all the elements below get shifted down as a result. Dirty hack in my opinion, but it does the job.
New problem now, however. :p
Opened the page in IE5.5. The absolutely positioned banner is centered on the page, but the page-container (and thus all other content) is not centered. It seemes margin: auto doesn't work in IE5. Are there any known hacks for this?