The fast will break by water going down the throat whilst gargling, even by mistake. This is important to remember whilst doing Wudhū or Ghusl. Gargle lightly. One Hadīth says:

وبالغ فى المضمضة الا ان تكون صائما

However, according to Imām Shāfi’ee’s R.A research, the fast will not break by such mistakes.

ذكره ابن العربى فى عارضة الاحوذى ص 189 / 2

Extracted from A Gift for Ramadhan, Page 54.

A Gift For Ramadhan – New 3rd Revised Edition