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I have drawn a cross section of a cylinder that contains a current density $J$. Around the cylinder there is a magnetic field $B$ because of Ampere's law. Inside the cylinder there is a whole. If you calculate with Ampere's law around a loop as I have drawn is it true that the magnetic field inside the hole has to be zero because no current passes through it?

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Ampere's law can only help if there is full cylindrical symmetry and a uniform current distribution. The trick for a general distribution is to consider this as two separate problems. First find the field with the empty space filled in by a convenient current distribution. Then add to this the magnetic field due to the current within the loop in the opposite direction (to cancel it). The total field anywhere will be the sum of these two fields. It will not usually be zero within the loop.

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