You're moving air, and the air can be cooled or heated to form the basis of energy transfer between the two chambers. So the system will work at some basic level.
However, it may not transfer much energy between the chambers in relation to the efficiency and energy used. If the chambers are adjacent, then it might work, but if they are separated and you have ventilation air hoses connecting them, then you'll need a decent fan to overcome the static pressure.
The Volumetric Heat Capacity is the amount of energy stored per unit volume per degree. For air, it's about 0.001, and for water it's 4.17. So, for the same volume and temperature increase, water will absorb over 4000 times as much energy - this makes it a much more efficient carrier. When using air, it will absorb a little energy and quickly rise in temperature to match it's surroundings, stopping the heat exchange early. This can be overcome by using a more powerful fan, but then this implies more cost.
So, water carries more energy per increase in temperature. But water is not an ideal medium here, since there is a real chance of freezing. Mixing in 30% glycol/glycerine will reduce the freezing point to well below the freezing point of water. Of course, the downside with water is the additional hardware required - tube, pump etc.
Given that your goal is to make a simple way to chill a second chamber to 10°C from a 1°C source, then I think you can probably stick with using air. Just be sure to have both in and out feeds between the chambers to get the best airflow.