German · Pronunciation Guide
dass TSIM-uh. The z is 'ts' and the -er ending softens into a dark 'uh'.
ZIM- is loud, -mer is not. The plural Zimmer is identical to the singular, so the article carries the whole burden of marking number — das Zimmer against die Zimmer — and nothing in the noun itself changes shape or rhythm.
Listen: das Zimmer
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Mein Zimmer ist klein.
My room is small.
Listen: Mein Zimmer ist klein.
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Mein Zimmer ist klein ends on klein, whose ei is the 'eye' diphthong. Mein shares that diphthong, so the sentence begins and ends with the same vowel, with the short clipped i of Zimmer sitting between them.
Reading z as an English z gives 'zimmer', which is exactly what the spelling suggests to an English eye. It is 'ts'. The ending is the second correction, and it is the same one that applies to Fenster, Mutter and Vater — the r dissolves into a vowel, so the word ends open rather than with a curled tongue.