We study the degree to which the cosmic microwave background (CMB) can be used to constrain primordial non-Gaussianity involving one tensor and two scalar fluctuations, focusing on the correlation of one polarization B mode with two temperature modes. In the simplest models of inflation, the tensor-scalar-scalar primordial bispectrum is nonvanishing and is of the same order in slow-roll parameters as the scalar-scalar-scalar bispectrumr. We calculate the {BTT} correlation arising from a primordial tensor-scalar-scalar bispectrum, and show that constraints from an experiment like CMB-Stage IV using this observable are more than an order of magnitude better than those on the same primordial coupling obtained from temperature measurements alone. We argue that B-mode non-Gaussianity opens up an as-yet-unexplored window into the early Universe, demonstrating that significant information on primordial physics remains to be harvested from CMB anisotropies.