This is a common question and a relatively simple answer. The only cold air produced by the compressor is to the freezer side. A fan in the freezer section cirulates the cold air to the fridge side. That cold air passes through a diverter door which you can see near the top of the fridge on the wall where it joins the freezer. It's usually called an air damper, at least here in the U.S. This damper is what your thermostat actually controls. It modulates the damper open and closed depending on demand for cold air in the fridge section.
The problem could be the thermostat has failed or it could be the electrical motor that drives the damper has failed. In either cae, an astute DIY'er could replace them, but it is a challenging job. If you call a repairman it can be fixed in short order, but be prepared for the cost...
Best of luck!