We report on the status of Z-Spec, including preliminary results of our first astronomical measurements. Z-Spec is a cryogenic, broadband, millimeter-wave grating spectrometer designed for molecular line surveys of galaxies, including carbon monoxide redshift measurements of high-redshift submillimeter sources. With an instantaneous bandwidth of 185-305 GHz, Z-Spec covers the entire 1 mm atmospheric transmission window with a resolving power of 200-400. The spectrometer employs the Waveguide Far-Infrared Spectrometer (WaFIRS) architecture, in which the light propagation is confined within a parallel-plate waveguide, resulting in a minimum mechanical envelope. Its array of 160 silicon-nitride micromesh bolometers is cooled to below 100 mK for background-limited performance. With its sensitivity, broad bandwidth, and compactness, Z-Spec serves as a prototype for a future far-IR spectrometer aboard a cold telescope in space. Z-Spec successfully demonstrated functionality with a partial array of detectors and warm electronics during a week-long engineering run at the Caltech Submillimeter Observatory in June, 2005. We describe the instrument performance evaluated at the telescope and in subsequent laboratory tests and compare these results with design specifications. Following several modifications we returned to the telescope in April, 2006. We present a preliminary astronomical spectrum and discuss our plans to improve sensitivity and throughput to achieve our ultimate science goals.