The winds will be strongest near coastal areas and there is a risk of flooding.
That warning will remain in place until 2am tomorrow morning.
"Widespread frosts are forecast on Thursday and Friday night and while the heaviest snow will fall over high ground in the north, there is a widespread risk of wintry showers including hail or sleet widely and possibly to lower levels into the weekend".
Elsewhere, very strong winds expected to reach 80mph will affect nearly the whole country on Wednesday night (17 January).
More than 50 schools and nurseries in the Highlands were closed due to the weather with over 3,000 children enjoying a snow day, while a number of schools in South Lanarkshire and Dumfries and Galloway were also affected.More news: Virat Kohli: A cut above the rest
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Storm Fionn, Storm Georgina, Storm David and Storm Friederike were all names being bandied about, but the Met Office has confirmed it is in fact Storm David.
The Atlantic coastal countries can expect winds of up to 80km/h and gusts up to 120km/h.
However, Storm Fionn hit Ireland much harder than the United Kingdom bringing with it gusts of 137 km/h at Mace Head on Tuesday.
The potential for more snow across the country will continue into the week.
Speaking on Morning Ireland on RTE Radio 1 Clara Finkele of Met Eireann said: "Winds will further moderate this afternoon ahead of the next depression early this evening, do check for further warnings on Met Eireann's website".
The wind is then expected to strengthen as it crosses the North Sea, with parts of the Netherlands and northern Germany seeing the strongest of the winds.