Some three "severe" warnings - denoting lives could be endangered by the floodwaters - remained active later Monday on two rivers in the English region.
Police said a man in his 60s died after entering the River Tawe, north of the Welsh city Swansea, but later clarified that the death was not "linked to the adverse weather".
There are 224 flood warnings in place, meaning flooding is expected and immediate action required.
A family is rescued from a property in Nantgarw, Wales yesterday.
South Wales Police said they had declared a major incident due to the flooding and severe weather.
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Wales bore the brunt of much of the weekend's rain and saw an unusual Red warning for Rain on Sunday in the south.
Tredegar in south-eastern Wales received 105 mm of rain in just a few hours, while rail services were stopped across South Wales and the line between Derby and Long Eaton was also suspended.
Dennis was the second major storm to hit the United Kingdom in a week, following the heavy rain, flooding and 90mph winds brought by Storm Ciara the previous weekend.
"The forecast is for very significant levels of rain, especially in the eastern valleys of South Wales", said Jeremy Parr, from Natural Resources Wales.
Storm Ciara's impact was heavily felt in West Yorkshire, where army personnel were deployed by the Ministry of Defence to assist people who have suffered damage from the storm and build flood defences in the streets.
Newly appointed environment secretary George Eustice said the government had done "everything that we can do with a significant sum of money" to combat increased flooding.
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Around 2,400 homes, farms and businesses nationally are without power this morning following Storm Dennis.
"It is important people are ready should the worst happen".
Meanwhile, British Airways and budget carrier EasyJet have confirmed they grounded flights.
On Saturday the storm claimed two victims in rough seas in Kent, where two bodies were pulled out of the water.
One man was found following a huge search operation off Margate after an early-morning distress call, while another was found at Herne Bay.
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