The Hubble Frontier Fields program combines the capabilities of the Hubble Space Telescope (HST) with the gravitational lensing of massive galaxy clusters to probe the distant universe to an unprecedented depth. Here, we present the results of the first combined HST and Spitzer observations of the cluster A-2744. We combine the full near-infrared data with ancillary optical images to search for gravitationally lensed high-redshift (z >~ 6) galaxies. We report the detection of 15 I 814 dropout candidates at z ~ 6-7 and one Y 105 dropout at z ~ 8 in a total survey area of 1.43 arcmin2 in the source plane. The predictions of our lens model also allow us to identify five multiply imaged systems lying at redshifts between z ~ 6 and z ~ 8. Thanks to constraints from the mass distribution in the cluster, we were able to estimate the effective survey volume corrected for completeness and magnification effects. This was in turn used to estimate the rest-frame ultraviolet luminosity function (LF) at z ~ 6-8. Our LF results are generally in agreement with the most recent blank field estimates, confirming the feasibility of surveys through lensing clusters. Although based on a shallower observations than what will be achieved in the final data set including the full Advanced Camera for Survey observations, the LF presented here goes down to M UV ~-18.5, corresponding to 0.2L sstarf at z ~ 7 with one identified object at M UV ~-15 thanks to the highly magnified survey areas. This early study forecasts the power of using massive galaxy clusters as cosmic telescopes and its complementarity to blank fields. Based on observations made with the NASA/ESA Hubble Space Telescope (HST), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs 13495 and 11689. Based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This work utilizes gravitational lensing models produced by PIs Ebeling, Merten, and Zitrin, and Sharon funded as part of the HST Frontier Fields program conducted by STScI. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. The lens models were obtained from the Mikulski Archive for Space Telescopes (MAST).