Carlow has been included in the warning which is valid from 6am on Tuesday, February 25 to 9am Wednesday, February 26.
Forecaster John Griffiths said between 5mm to 10mm could fall on the River Severn's source, the Welsh hills, throughout Wednesday, with other parts of the United Kingdom seeing up to 2mm.
As further heavy rain is expected to cross the North West over the next 24 hours, the Environment Agency is warning that flood risk continues with the impacts of stormy weather continuing to be felt.
The Met Office has issued a yellow weather alert for ice and snow across a large part of the UK later this evening and into Wednesday.
Parts of the United Kingdom could see temperatures of minus 10C as forecasters warn of snow and ice over the next 24 hours as Britain prepares to freeze.
After heavy snowfall in parts of Scotland and Northern Ireland, experts predict snow could even fall in London and southern England today and tomorrow.
The national forecaster added that "some snow accumulations will occur".
Dan Suri, chief forecaster at the Met Office, said: "A relatively deep area of low-pressure system on Monday provides a continuation of the extremely unsettled period the United Kingdom has endured".
A separate weather warning for snow stretches from South Wales and the Bristol Channel all the way up through northern England and Scotland.
As per reports, Storm Dennis also affected the northwest region of France as approximately 60,000 people suffered power cuts, while rails and traffic were also disrupted in the region.
"With this volume of snow on higher routes, we expect this to become a significant event".
Forecaster Marco Petagna said Saturday would bring a respite from the rain in some areas but there was a warning for high winds in the north and another band of stormy weather would sweep in from the South West on Saturday night.
"Other modes of transport - ferries, trains and flights may also be disrupted, so we ask people check with their operators before setting out". Plan your journey in advance, allow extra time and make sure that your vehicle is adequately prepared.