The hydrohelium cation, HeH(+), serves as an important benchmark for ab initio calculations that take into account non-adiabatic, relativistic, and quantum electrodynamic effects. Such calculations are capable of predicting molecular transitions to an accuracy of ~300 MHz or less. However, in order to continue to push the boundaries on these calculations, new measurements of these transitions are required. Here we measure seven rovibrational transitions in the fundamental vibrational band to a precision of ~1 MHz using the technique of Noise Immune Cavity Enhanced Optical Heterodyne Velocity Modulation Spectroscopy. These newly measured transitions are included in a fit to the rotation-vibration term values to derive refined spectroscopic constants in the v = 0 and v = 1 vibrational states, as well as to calculate rotation-vibration energy levels with high precision.