Thanks for the reply. It didn't work (well, I couldn't make it work)
However, I found another solution which works.
From the source code of apacite.sty I got the idea that apacite must
read a "global" langauge parameter. It shouldn't be parsed via the
usepackage call.
So, first of all, there is no need to hack the apacite.sty with this
line:\APAC@iflang{danish}{\gdef\APAC@apcfile{danish.apc}}
as it is only used (as far as I can see) if no language is set at all.
Secondly, the .apc-file must have the same name as the "global"
language, that is danish.apc (and not dansk.apc)!!
Thirdly, in order to work, the call to babel must be with just one
language option. That is, \usepackage[danish,english]{babel} doens't
work, however, \usepackage[danish]{babel} does.
Thus, the preamble looks like this:
\documentclass[a4paper,12pt]{article}
\usepackage[ansinew]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[danish]{babel}
\usepackage{apacite}
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