We present the performance and radiation tolerance of an analog application-specified integrated circuit (ASIC) developed for the engineering model of X-ray charge-coupled device (CCD) camera onboard the next Japanese astronomical satellite. The ASIC has four identical channels and each of them equips a pre-amplifier and two $\Delta\Sigma$ analog-to-digital converters. The 3 mm square bare chip has been packaged into the 15 mm square quad flat plastic. The front-end electronics test proved its power consumption to be 71 mW for the whole chip at the readout pixel rate of 20 kHz. The equivalent input noise was $32.8\pm 0.3\ \mu{\rm V}$ and the integrated non-linearity was 0.2% throughout its dynamic range of $\pm$ 20 mV. At the integrated test with an X-ray CCD, we put an identical signal to all of four channels and took the average of their outputs. Then the noise performance improved to be $17.9\pm 0.3\ \mu{\rm V}$ and the energy resolution of Mn ${\rm K}\alpha$ line from $^{55}{\rm Fe}$ reached down to 135 $\pm$ 3 eV (full-width at half-maximum). In order to investigate the radiation tolerance against the total ionizing dose effect, the ASIC was irradiated with 200 MeV proton beam at HIMAC/NIRS in Japan. There was no significant degradation of gain and noise performance until the absorbed dose amounted up to 15 krd, which corresponds to $>$ 10 years in the planned low earth orbit (LEO). Although the noise suddenly increased at $>$ 15 krd, there was no significant increase of the current in the chip and the performance recovered after the annealing at the room temperature for three months. This suggests that the degradation during the test was caused by temporal charge trapping near or at the interface of ${\rm SiO}_{2}$ and Si bulk. Considering that the typical mission lifetime of X-ray astronomical satellites is $\leq$10 years, we proved that our ASIC has sufficient radiation tolerance for the use in the LEO.