This paper reports the discovery of a Very Low Luminosity Object (VeLLO) in the "starless" dense core L328, using the Spitzer Space Telescope and ground based observations from near-infrared to millimeter wavelengths. The Spitzer 8 micron image indicates that L328 consists of three subcores of which the smallest one may harbor a source, L328-IRS while two other subcores remain starless. L328-IRS is a Class 0 protostar according to its bolometric temperature (44 K) and the high fraction ~72 % of

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... its luminosity emitted at sub-millimeter wavelengths. Its inferred "internal luminosity" (0.04 - 0.06 Lsun) using a radiative transfer model under the most plausible assumption of its distance as 200 pc is much fainter than for a typical protostar, and even fainter than other VeLLOs studied previously. Note, however, that its inferred luminosity may be uncertain by a factor of 2-3 if we consider two extreme values of the distance of L328-IRS (125 or 310 pc). Low angular resolution observations of CO do not show any clear evidence of a molecular outflow activity. But broad line widths toward L328, and Spitzer and near-infrared images showing nebulosity possibly tracing an outflow cavity, strongly suggest the existence of outflow activity. Provided that an envelope of at most ~0.1 Msunis the only mass accretion reservoir for L328-IRS, and the star formation efficiency is close to the canonical value ~30%, L328-IRS has not yet accreted more than 0.05 Msun. At the assumed distance of 200 pc, L328-IRS is destined to be a brown dwarf.