Without knowing the age of the item it is difficult. But given your description the most likely cause by far is : Failed fridge section thermostat. To check this, turn the control knob fully clockwise (coldest position) and then slowly back to '0'. Listen and feel for a positive 'click' from the unit. If you have a 'click' it is OK, if not you need to replace it. It is within the scope of DIY normally. You do not need to obtain the part from the manufacturer, this is a 'one-size-fits-all' part, check around at local likely shops, or internet sites.... Cost should be around 10 to £15. If your unit is over say 8 years old you most likely are looking for a Ranco VT9 or Danfoss 03.
Trevor
June 2007