The new transient IGR J16358-4726 was discovered on 2003 March 19 with INTEGRAL. We detected the source serendipitously during our 2003 March 24 observation of SGR 1627 - 41 with the CHANDRA X-ray Observatory at the 1.7 x 10 - 1 0 ergs s - 1 cm - 2 flux level (2-10 keV) with a very high absorption column (N H =3.3x10 2 3 cm - 2 ) and a hard power law spectrum of photon index 0.5(1). We discovered a very strong flux modulation with a period of 5880(50) s and peak-to-peak pulse fraction of 70(6) % (2-10 keV), clearly visible in the x-ray data. The nature, however, of IGR 16357-4726 remains unresolved. Most likely, we are looking at a galactic binary neutron star system as evidenced by the detection of rest frame fluorescence line emission from neutral Fe K (6.4 keV) in the Chandra spectrum. If the detected modulation is a spin period, this transient is a new kind of object, belonging to a class of very hard transients recently revealed with the unique INTEGRAL capabilities. The only neutron star systems known with similar spin periods are low luminosity persistent wind-fed pulsars, which exhibit very soft X-ray spectra. If this is an orbital period, then the system could be a compact Low Mass X-ray Binary.