In all known Group 5 transition-metal dichalcogenide monolayers (MLs), the metal centers carry a spin, and their ground-state phases are either metallic or semiconducting with indirect band gaps. Here, on grounds of first-principles calculations, we report that the Haeckelite polytypes 1S-NbX₂ (X=S, Se, Te) are diamagnetic direct-band-gap semiconductors even though the Nb atoms are in the 4+ oxidation state. In contrast, 1S-VX₂ MLs are antiferromagnetically coupled indirect-band-gap semiconductors. The 1S phases are thermodynamically and dynamically stable but of slightly higher energy than their 1H and 1T ML counterparts. 1S-NbX₂ MLs are excellent candidates for optoelectronic applications owing to their small band gaps (between 0.5 and 1 eV). Moreover, 1S-NbS₂ shows a particularly high hole mobility of 2.68×10³ cm² V⁻¹ s⁻¹, which is significantly higher than that of MoS₂ and comparable to that of WSe₂.