Current weather, local time, UTC offset and DST rules for 400 cities as static JSON.
World Time & Weather API is a free Weather API that requires no API key, is served over HTTPS and supports CORS, so it can be called from the browser.
Local time, UTC offset, DST dates and current weather for 400 cities as plain JSON. No signup, no rate limit.
Details from worldtimeweather.com.

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Yes. The World Time & Weather API is listed in the Weather category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The World Time & Weather API requires no authentication, so you can start making requests without signing up.
Yes. The World Time & Weather API supports CORS, so front-end JavaScript can call it directly.
Yes. The World Time & Weather API is served over HTTPS, so it's safe to use in production.