  We present the identification of a bright submm source, SMMJ163555.5+661300,
detected in the lensing cluster Abell2218, for which we have accurately
determined the position using observations from the Submillimeter Array (SMA).
The identified optical counterpart has a spectroscopic redshift of
z=4.044+-0.001 if we attribute the single emission line detected at
lambda=6140AA to Lyman-alpha. This redshift identification is in good agreement
with the optical/near-infrared photometric redshift as well as the submm flux
ratio S_450/S_850~1.6, the radio-submm flux ratio S_1.4/S_850 < 0.004, and the
24um to 850um flux ratio S_24/S_850 < 0.005. Correcting for the gravitational
lensing amplification of ~5.5, we find that the source has a far-infrared
luminosity of 1.3x10^12 Lsun, which implies a star formation rate of 230
Msun/yr. This makes it the lowest-luminosity SMG known at z>4 to date. Previous
CO(4-3) emission line obserations yielded a non-detection, for which we derived
an upper limit of the CO line luminosity of L'_CO = 0.3x10^10 K km/s/pc^2,
which is not inconsistent with the L'_CO - L_FIR relation for starburst
galaxies. The best fit model to the optical and near-infrared photometry give a
stellar population with an age of 1.4 Gyr and a stellar mass of 1.6x10^10 Msun.
The optical morphology is compact and in the source plane the galaxy has an
extent of ~6kpc x 3kpc with individual star forming knots of <500 pc in size.
J163556 is not resolved in the SMA data and we place a strict upper limit on
the size of the starburst region of 8kpc x 3kpc, which implies a lower limit on
the star formation rate surface density of 12 Msun/yr/kpc^2. The redshift of
J163556 extends the redshift distribution of faint, lensed SMGs, and we find no
evidence that these have a different redshift distribution than bright SMGs.
