A coin wrapping machine includes a discriminator for discriminating whether or not a coin is acceptable, a denomination of the coin when the coin is acceptable and whether or not the denomination of the coin coincides with that specified and counting coins of the specified denomination, a coin stacking device for stacking coins of the denomination to be wrapped, a reference stacked coin height data memory for storing reference stacked coin height data for each denomination of coins stacked by the coin stacking device, and a coin number discrepancy detecting device including a comparator for comparing a height of stacked coins and the reference stacked coin height data for each denomination stored in the reference stacked coin height data memory, the discriminator being made responsive to inclusion in coins of the specified denomination of new issue coins and past issue circulating coins for discriminating whether each coin is a new issue coin or a past issue circulating coin, determined based on a count made by the discriminator.