Over higher ground there could be between two and four inches of snow overnight and into tomorrow morning.
Forecasters are also warning that there could be accumulations of snow at lower levels, and these could lead to travel problems during Thursday's morning rush hour.
The weather warning states: "Outbreaks of rain and sleet, with snow over hills will spread east during Wednesday, before increasingly turning to snow in places.
"This seems likely to happen during Wednesday evening across southern areas of Scotland, perhaps including parts of the central belt.
"The trend to wintry weather then extends into parts of northern England during the rest of Wednesday night, clearing from Yorkshire during the morning.
"At this stage, it looks as if the heaviest snow will affect parts of southern Scotland (for which a separate Amber warning is in force) and upland areas of northern England, including cross-Pennine routes, where 3 to 6 cm may fall in a number of areas, with possibly as much as 10 cm locally, mainly above elevations of 300 metres.
"Please be aware of the risk of disruption to travel, and take appropriate precautions before undertaking journeys. It will also be worth keeping fully up to date with the latest forecasts - details of amounts and areas affected may well change."