The growth of graphene by catalytic decomposition of ethylene on Cu(111) in an ultra-high vacuum system was investigated with low energy electron diffraction, low energy electron microscopy, and atomic force microscopy. Attempts to form a graphene overlayer using ethylene at pressures as high as 10 mTorr and substrate temperatures as high as 900 $^\circ$C resulted in almost no graphene growth. By using an argon overpressure, the growth of epitaxial graphene on Cu(111) was achieved. The suppression of graphene growth without the use of an argon overpressure is attributed to Cu sublimation at elevated temperatures. During the initial stages of growth, a random distribution of rounded graphene islands is observed. The predominant rotational orientation of the islands is within $\pm1^\circ$ of the Cu(111) substrate lattice.