  We experimentally study a gas of quantum degenerate $^{87}$Rb atoms
throughout the full dimensional crossover, from a one-dimensional (1D) system
exhibiting phase fluctuations consistent with 1D theory to a three-dimensional
(3D) phase-coherent system, thereby smoothly interpolating between these
distinct, well-understood regimes. Using a hybrid trapping architecture
combining an atom chip with a printed circuit board, we continuously adjust the
system's dimensionality over a wide range while measuring the phase
fluctuations through the power spectrum of density ripples in time-of-flight
expansion. Our measurements confirm that the chemical potential $\mu$ controls
the departure of the system from 3D and that the fluctuations are dependent on
both $\mu$ and the temperature $T$. Through a rigorous study we quantitatively
observe how inside the crossover the dependence on $T$ gradually disappears as
the system becomes 3D. Throughout the entire crossover the fluctuations are
shown to be determined by the relative occupation of 1D axial collective
excitations.
