Yes , Conroe is a 64-bit processor.
In one mode, legacy mode, the mode in which most x86-64 processors still run today, it's a 100% 32-bit processor. In this mode, it has no added 64-bit functionality nor is it a 32-bit processor that can run 64-bit. It's exactly the same as all previous 32-bit x86 processors. It has no additional tricks and certainly can't run 64-bit code either.
In another mode, long mode, it's a new 64-bit processor. In this mode it's a different processor and not fully backwards compatible with 32-bit x86. The system interface is different so it's not compatible with old system level instructions and interrupts. It does however have a layer that converts segment registers and a 32-bit address into a 64-bit address. So this way it can execute old application level 32-bit instructions. This provides the ability, for a 64-bit OS, to also run old 32-bit applications.