Café Moskau sits close to the Karl Marx Monument and has served coffee in roughly this form since the Karl-Marx-Stadt days, when a Café Moskau was a fixture of every East German city centre. The name has outlasted the ideology, which feels appropriate in Chemnitz. It's a good place for coffee and cake after the monument and the nearby museums — the Baumkuchen and the Pflaumenkuchen are both worth ordering.
The atmosphere is unhurried and the prices are low. It fills up on Saturday mornings with people doing the market and browsing the central streets. A reliable, unglamorous, very Saxon café.