This is what the closest equivalent in the UK is somewhat euphemistically called - however they are forbidden by law from selling alcohol so it is not the same thing. There is one in our local shopping precinct just called 'Amusement's' (including the grocer's apostrophe) and the clientèle is similar. Maybe there is a better US term for it. Google it for more explanation - they also don't always tend to have exclusively gaming machines, but they mostly do.

or: non-stop gambling lounge

establishment- mozno sa vam podari najst nieco podobne, mozno sa vam to bude pozdavat.
Vysvetlenie som uz napisala v diskusii, ale ako priklada by som uviedla co som nasla pomocou internetu.
http://en.wikipedia.org/wiki/Indiana_Live_Casino
"Indiana Live! Casino- is a Las Vegas style gaming establishment in Shelbyville, Indiana, that offers non-stop gaming action, award-winning food and wide array of entertainment options all under one roof".
Tu (USA/MI ) su ludia (urcita skupina) velmi citlivi na vyraz casino; potvrdene nekonecnymi debatami za poslednych 15-17 rokov nasho mesta a okolia, hoci herne im nevadia.

I think 'den' (CZ 'nora', 'doupě') conveys the idea of something seedy - it is usually, but not exclusively, accompanied by the word 'illegal'
http://www.thefreedictionary.com/gambling den

As I said above, 'amusement arcades' are forbidden from selling alcohol, but it is the closest we have, legally, in the UK. However, most pubs have a fruit machine (or 'bandit') in the corner, but I don't think there is any special term for that corner.

The Prague post used 'gambling parlours' here http://www.praguepost.com/news/1739-state-slammed-over-rise-... but IMO that sounds too luxurious ...

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Note added at 5 hrs (2010-11-29 16:59:16 GMT)
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I know you'r not supposed to add more, but 'seedy gambling joint' sprang to mind - quite a few google hits.

It isn't an exact translation, of course, but it is a similar concept for a place with games, including slot machines, and alcohol is served, unlike at regular amusement arcades.
It also hints at the seedy aspect of a herna bar.

"After-hours" is the term commonly used in North America.