An image data coding apparatus performs first and second coding processes. A controller outputs first and second control signals. An orthogonal transform circuit performs orthogonal transform on image data of one screen and generates an transformed output. A quantizing circuit quantizes the transformed output with a predetermined temporary first quantization width upon reception of the first control signal and generates a first quantized output, and quantizes the transformed output with an optimal second quantization width upon reception of the second control signal and generates a second quantized output. An entropy coding circuit codes the first quantized output and generates a first coded output, and codes the second quantized output and generates a second coded output. An amount-of-codes calculator calculates the amount of codes of the first coded output for each block and calculates the amount of codes of one screen based on the amount of codes for each block. An assigned-amount-of-codes determining circuit determines the amount of codes assigned to each block based on the amount of codes for each block and the amount of codes of the one screen, and the target amount of codes. A quantization width predicting circuit predicts the second quantization width based on the amount of codes of the one screen. A coding stop circuit supplies a coding stop signal to the entropy coding circuit when the amount of codes of the second coded output exceeds the assigned amount of codes.