The Micro-X sounding rocket experiment will deploy an imaging transition-edge-sensor (TES) microcalorimeter spectrometer to observe astrophysical sources in the 0.2 - 3.0 keV band. The instrument has been designed at a systems level, and the first items of flight hardware are presently being built. In the first flight, planned for January 2011, the spectrometer will observe a recently discovered Silicon knot in the Puppis-A supernova remnant. Here we describe the design of the Micro-X science instrument, focusing on the instrument's detector and detector assembly. The current design of the 2-dimensional spectrometer array contains 128 close-packed pixels with a pitch of 600 |im. The conically approximated Wolter-1 mirror will map each of these pixels to a 0.95 arcmin region on the sky; the field of view will be 11.4 arcmin. Targeted energy resolution of the TESs is about 2 eV over the fiill observing band. A SQUID time-division multiplexer (TDM) will read out the array. The detector time constants will be engineered to approximately 2 ms to match the TDM, which samples each pixel at 32.6 kHz, limited only by the telemetry system of the rocket. The detector array and two SQUID stages of the TDM readout system are accommodated in a lightweight Mg enclosure, which is mounted to the 50 mK stage of an adiabatic demagnetization refrigerator. A third SQUID amplification stage is located on the 1.6 K liquid He stage of the cryostat. An on-board 55-Fe source will fluoresce a Catarget, providing 3.69 and 4.01 keV calibration lines that will not interfere with the scientifically interesting energy band.