  We report a rectangular charge density wave (CDW) phase in strained
1T-VSe$_2$ thin films synthesized by molecular beam epitaxy on c-sapphire
substrates. The observed CDW structure exhibits an unconventional rectangular
4a{\times}{\sqrt{3a}} periodicity, as opposed to the previously reported
hexagonal $4a\times4a$ structure in bulk crystals and exfoliated thin layered
samples. Tunneling spectroscopy shows a strong modulation of the local density
of states of the same $4a\times\sqrt{3}a$ CDW periodicity and an energy gap of
$2\Delta_{CDW}=(9.1\pm0.1)$ meV. The CDW energy gap evolves into a full gap at
temperatures below 500 mK, indicating a transition to an insulating phase at
ultra-low temperatures. First-principles calculations confirm the stability of
both $4a\times4a$ and $4a\times\sqrt{3}a$ structures arising from soft modes in
the phonon dispersion. The unconventional structure becomes preferred in the
presence of strain, in agreement with experimental findings.
