1. A carbon-fiber-precursor acryl fiber bundle, comprising: a polyacrylonitrile-based copolymer comprising from 95 to 99 mol % of an acrylonitrile unit and from 1 to 5 mol % of a hydroxyalkyl (meth)acrylate unit, wherein the fiber bundle has a single-fiber fineness of from 1.5 dtex to 3.0 dtex and a roundness of 0.9 or less in a cross-section shape perpendicular to a fiber axis of the single fiber, wherein the roundness is a value determined using equation (1): roundness=4πS/L 2 , wherein S is a cross-sectional area of the single fiber and L is a circumferential length of the single fiber, and S and L are obtained by observing, under an SEM, the cross-section of the single fiber perpendicular to the fiber axis of the single fiber and analyzing the obtained image, and wherein when a constant velocity temperature rising exothermic curve is from 30° C. to 450° C. measured at a temperature rising rate of 10° C./minute in air flow at 100 ml/minute at 30° C. and 0.10 MPa using a heat flux type differential scanning calorimeter; a heat quantity Ja obtained by integrating an exothermic velocity from 230° C. to 260° C. of the constant velocity temperature rising exothermic curve is from 100 kJ/kg to 250 kJ/kg; and a heat quantity Jb obtained by integrating an exothermic velocity from 260° C. to 290° C. is from 550 kJ/kg to 1050 kJ/kg.