The answer is "it depends." Here are some of the factors on which it depends:
- The thickness of the ice.
Ice is a mediocre conductor of heat, about the same as rock. A thick layer of ice somewhat insulates the upper surface of the ice from the ~0 °C water just below the ice. A thin layer of ice, the ice will be at ~0 °C.
- The average wind speed.
Thermal conductivity is the dominant form of heat transfer in windy and breezy conditions. Radiative cooling dominates when the wind is very calm.
- The percent cloudiness, and the altitude of the clouds.
The percent cloudiness is a key driver in the night sky temperature (which is very important when radiative cooling dominates over thermal conductivity). Low clouds are warmer than higher clouds, so the height of the bottom of the clouds is important when clouds are present.
- The relative humidity.
Humidity is another key driver of the night sky temperature.
- The time of day (or time of night, in this case).
Air has a much lower heat capacity than does ice. Ice holds the warmth of the day much longer than does air.
- The amount of snow that is atop the ice.
Even a few centimeters of snow acts as a rather nice blanket. A meter of snow makes for a very, very nice blanket.
- What you mean by "air temperature -10 degrees Celsius."
The standard is to measure the temperature at 1.5 meters above the surface. I'll assume the standard.
The question eliminates one of the variables; there is no wind. It doesn't truly address the thickness of ice; the question stipulates ice of "some nominal thickness." There's a big difference between ice that is safe to walk on versus ice that is safe to drive a four wheel truck on versus ice that is safe to drive a fully-loaded eighteen wheeler. It doesn't touch on the other variables at all.
To illustrate, suppose the ice is covered with a meter of snow, and it's snowing. That meter of snow will make for a very nice blanket. That it's snowing means its cloudy and that the humidity is high. The surface of the ice will be just a bit below 0 °C.
On the other hand, suppose it's just before sunrise, the ice has no snow cover and is thick enough to support a four wheeled vehicle (~ 40 cm thick), the sky is perfectly clear, and the air is extremely calm and extremely dry. Now the surface of the ice can be well below -10 °C.
A related phenomenon occurs on clear, calm, and dry nights when the low is to 4 °C or so. You might well see frost on the roof of your house and on your grass, and if you have a pet with an outdoor water bowl, the water might well be covered by a thin layer of ice. Yet the temperature never dropped below freezing. The reason is radiative cooling dominates over thermal conduction on those clear, calm, and dry nights.