The growth and elementary properties of $p$-type ${\text{Bi}}_{2}{\text{Se}}_{3}$ single crystals are reported. Based on a hypothesis about the defect chemistry of ${\text{Bi}}_{2}{\text{Se}}_{3}$, the $p$-type behavior has been induced through low-level substitutions (1% or less) of Ca for Bi. Scanning tunneling microscopy is employed to image the defects and establish their charge. Tunneling and angle-resolved photoemission spectra show that the Fermi level has been lowered into the valence band by about 400 meV in ${\text{Bi}}_{1.98}{\text{Ca}}_{0.02}{\text{Se}}_{3}$ relative to the $n$-type material. $p$-type single crystals with $ab$-plane Seebeck coefficients of $+180\text{ }\ensuremath{\mu}\text{V}/\text{K}$ at room temperature are reported. These crystals show an anomalous peak in the Seebeck coefficient at low temperatures, reaching $+120\text{ }\ensuremath{\mu}\text{V}\text{ }{\text{K}}^{\ensuremath{-}1}$ at 7 K, giving them a high thermoelectric power factor at low temperatures. In addition to its interesting thermoelectric properties, $p$-type ${\text{Bi}}_{2}{\text{Se}}_{3}$ is of substantial interest for studies of technologies and phenomena proposed for topological insulators.