der Ausspruch
saying
A noteworthy, often quoted statement or well-known sentence by a person.
Der berühmte Ausspruch „Der Würfel ist gefallen“ wird Julius Cäsar zugeschrieben.
The famous saying 'The die is cast' is attributed to Julius Caesar.

saying / pronouncement
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saying
A noteworthy, often quoted statement or well-known sentence by a person.
Der berühmte Ausspruch „Der Würfel ist gefallen“ wird Julius Cäsar zugeschrieben.
The famous saying 'The die is cast' is attributed to Julius Caesar.

'Der Ausspruch' (the saying / pronouncement) is the nominalization of the verb 'aussprechen' (to pronounce / to say). It therefore refers to something that was said or pronounced publicly or formally.
Der Zeuge weigerte sich, den Namen des Täters auszusprechen.
The witness refused to say the perpetrator's name.
The meaning depends strongly on the context. If the topic is history, literature, or a famous person, it usually means a quoted statement. In connection with 'Gericht' (court), 'Richter' (judge), or 'Urteil' (judgment), it refers to the legal decision.
„Ich bin ein Berliner“ ist ein berühmter Ausspruch von John F. Kennedy.
'Ich bin ein Berliner' is a famous saying by John F. Kennedy.
When referring to a quoted statement, 'Ausspruch' (saying) is often combined with adjectives such as 'berühmt' (famous), 'bekannt' (well-known), or 'historisch' (historic). You can also say 'einen Ausspruch machen' (to make a statement / saying).
Auf ihrer Reise machte sie einige denkwürdige Aussprüche.
On her trip, she made several memorable remarks.
In the meaning 'Urteilsverkündung' (pronouncement of judgment), 'Ausspruch' (pronouncement) is a technical term, specifically from legal language. In everyday life, you will rarely hear this word in this context.
Der schriftliche Ausspruch des Urteils wird den Anwälten zugestellt.
The written pronouncement of the judgment will be sent to the lawyers.
word
'das Wort' and 'der Ausspruch' can both mean something that has been said. 'Wort' is very general and can mean a short utterance or several spoken words. 'Ausspruch' is more specific: it usually means a special sentence that someone said and that people remember or quote. An 'Ausspruch' often sounds meaningful, wise, or typical of a person. For normal everyday utterances, 'Wort' is more natural; for a well-known or noteworthy sentence, 'Ausspruch' fits better.

An diesen Ausspruch seiner Großmutter denkt Paul noch oft.
Paul still often thinks about this saying of his grandmother's.
Sag bitte ein paar Worte zur Begrüßung der Gäste.
Please say a few words to welcome the guests.