A small calorimeter for massic heat capacity c(v) measurements in the criticl region of fluids was developed. Its main part is a horizontal capillary tube. The influence of gravity is reduced, and the distance for the transfer of heat into the fluid is shortened with the small diameter of the tube. The calibration of the calorimeter, the experimental procedure, and the results are described. The influence of vertical and radial profiles of temperature and density and the experimental uncertainty are discussed. Results for c(v) of CO2 at nearly critical density are shown.