  The microscopic mechanism of Cooper pairing in a superconductor leaves its
fingerprint on the symmetry of the order parameter. UTe$_2$ has been inferred
to have a multi-component order parameter that entails exotic effects like time
reversal symmetry breaking. However, recent experimental observations in
newer-generation samples have raised questions about this interpretation,
pointing to the need for a direct experimental probe of the order parameter
symmetry. Here, we use pulse-echo ultrasound to measure the elastic moduli of
UTe$_2$ in samples that exhibit both one and two superconducting transitions.
We demonstrate the absence of thermodynamic discontinuities in the shear
elastic moduli of both single- and double-transition samples, providing direct
evidence that UTe$_2$ has a single-component superconducting order parameter.
We further show that superconductivity is highly sensitive to compression
strain along the $a$ and $c$ axes, but insensitive to strain along the $b$
axis. This leads us to suggest a single-component, odd-parity order parameter
-- specifically the B$_{2u}$ order parameter -- as most compatible with our
data.
