Programming and program-verification methods for low-voltage flash memories using the Fowler-Nordheim tunneling mechanism for both programming and erasure are described. In these memories, a great many memory cells on a selected word line, such as 512-bytes worth of cells, are programmed at the same time for high-speed programming. The bit-by-bit programming/verification method can precisely control threshold-voltage deviation of programmed memory cells on the selected word line for low voltage operation. By using an internal program-end detection circuit, the completion of program mode can be checked for in one clock cycle, without reading out 512-bytes of data from the memory chip to the external controller. Moreover, the variable pulse-width programming method reduces the total number of verifications.