This paper addresses a problem of finding an optimal dynamic quantizer for nonlinear control subject to discrete-valued signal constraints, i.e., to the condition that some signals must take a value on a discrete and countable set at each time instant. The quantizers to be studied are in the form of a nonlinear difference equation which maps continuous-valued signals into discrete-valued ones. They are evaluated by a performance index expressing the difference between the resulting quantized system and the unquantized system, in terms of the input-output relation. In this paper, we present a closed-form solution, which globally minimizes the performance index. This result shows the performance limitation of a general class of dynamic quantizers. In addition to this, some results on the structure and the stability are given in order to clarify the mechanism of the best dynamic quantization in nonlinear control systems.