The Pool Two match was switched to Dunbar in East Lothian due to the pitch being unplayable at Bridgehaugh.
In the first half the Scottish side built up a 10-8 lead thanks to tries from centre Byron McGuigan and prop Mike McDonald. Bedwas' points in that first 40 minutes came through a try from winger Rory Budina and then a penalty by scrum-half Ethan Davies. After the interval, six more tries were scored by the two sides as things opened up. County grabbed two through winger Ross Aitken and second-row Craig Deacons, with stand-off Stuart Edwards kicking one conversion.
However, the four tries scored by the visitors through hooker Elliot Dee, replacement James Thomas, back-row Deri Mattravers and centre Ross Wardle – allied to Davies's two conversions – put the game out of County's reach. Both teams have won two from three so far ahead of the rematch in Wales next weekend.
Dundee HSFP came agonisingly close to securing their first victory in the tournament, but two late penalties from the hosts saw them go down 20-17 at Carmarthen. The Scots had led 17-14 in Wales before the late drama but have now lost all three matches in Pool Three.
In the RBS Regional Cup, several matches fell foul of the weather, but Heriot's defeated Murrayfield Wanderers 34-24 in the Edinburgh section at Goldenacre.
The RBS Premiership side were given a few scares along the way by the men from the Championship. Skipper Graham Wilson led the way for the home side with two tries while back-row Jamie Syme also went over for two.
Centre Max Learmonth and hooker Stewart Mustard scored Heriot's other two tries with Wilson slotting two conversions.
Alex Cox scored two tries for Murrayfield Wanderers, with Chris Hill and Ally Borland also going over and Chris Beattie kicking two conversions.
Elsewhere in the Edinburgh section, Biggar won 39-12 at Haddington.
In the RBS West Regional Cup, GHA travelled to Ardrossan Accies and earned a 37-3 victory. Greenock Wanderers defeated Whitecraigs 25-20 at Fort Matilda.
All matches in the RBS Borders and RBS Caledonia Regional Cups were off along with the other matches in the Edinburgh and West sections.