The rotational spectrum of silyl isocyanide (SiH$_3$NC), an isomer of the well studied silyl cyanide (SiH$_3$CN), has been detected in the laboratory in a supersonic molecular beam, and the identification was confirmed by observations of the corresponding rotational transitions in the rare isotopic species SiH$_3$$^{15}$NC and SiH$_3$N$^{13}$C. Spectroscopic constants derived from 19 transitions between $11 - 35$~GHz in the three lowest harmonically related rotational transitions in the $K = 0 ~{rm{and}}~1$ ladders of the normal isotopic species including the nitrogen nuclear quadrupole hyperfine constant, allow the principal astronomical transitions of SiH$_3$NC to be calculated to an uncertainty of about 4~km~s$^{-1}$ in equivalent radial velocity, or within the FWHM of narrow spectral features in the inner region of IRC+10216 near 200~GHz. The concentration of SiH$_3$NC in our molecular beam is three times less than SiH$_3$CN, or about the same as the corresponding ratio of the isomeric pair SiNC and SiCN produced under similar conditions. Silyl isocyanide is an excellent candidate for astronomical detection, because the spectroscopic and chemical properties are very similar to SiH$_3$CN which was recently identified in the circumstellar envelope of IRC+10216 by citet{cernicharo_discovery_2017} and of SiNC and SiCN in the same source.
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