We apply small-angle neutron scattering (SANS) to study the effect of pressure on micelle structure in a solution of 1% by weight pentaethylene glycol mono-n-dodecyl ether (C12E5) in D2O at 20 degreesC and pressures up to similar to 3000 bar. At ambient pressure, the structure is a network of branched, semiflexible, cylindrical micelles with the branch points comprised of three-armed junctions. Our SANS results reveal that pressure induces a phase transition from this network of threadlike micelles to hexagonally ordered bundles of cylindrical micelles. Using geometric packing constraints for three-arm junctions and cylinders, we show that the formation of three-arm junctions becomes increasingly unfavorable with increasing pressure due to the compression of the micelle hydrophobic core, and as such, the network becomes unstable at pressures close to those observed in our SANS experiments. We also measured the temperature dependence of the transition pressure and find that it follows the pressure-temperature freezing curves for liquid n-alkanes of comparable hydrocarbon chain length. These observations lead us to propose that the phase transition is related to a loss of flexibility or conformational entropy of the C12E5 micelles upon the pressure-induced freezing of the micelle hydrophobic core to form an amorphous solid. The formation of hexagonally ordered bundles of cylindrical micelles follows as attractive van der Waals forces between the micelles are not offset by the loss of repulsive undulation forces arising from the fluidity of the hydrophobic core.