We present new mid- to far-infrared images of the two dwarf compact elliptical galaxies that are satellites of M31 , NGC 185 and NGC 147 , obtained with the Spitzer Space Telescope . Spitzer ’ s high sensitivity and spatial resolution enable us for the first time to look directly into the detailed spatial structure and properties of the dust in these systems . The images of NGC 185 at 8 and 24 micron display a mixed morphology characterized by a shell-like diffuse emission region surrounding a central concentration of more intense infrared emission . The lower resolution images at longer wavelengths show the same spatial distribution within the central 50″ but beyond this radius , the 160 \mu m emission is more extended than that at 24 and 70 \mu m. On the other hand , the dwarf galaxy NGC 147 located only a small distance away from NGC 185 shows no significant infrared emission beyond 24 micron and therefore its diffuse infrared emission is mainly stellar in origin . For NGC 185 , the derived dust mass based on the best fit to the spectral energy distribution is 1.9 \times 10 ^ { 3 } M _ { \odot } , implying a gas mass of 3.0 \times 10 ^ { 5 } M _ { \odot } . These values are in agreement with those previously estimated from infrared as well as CO and HI observations and are consistent with the predicted mass return from dying stars based on the last burst of star formation 1 \times 10 ^ { 9 }  yr ago . Based on the 70 to 160 \mu m flux density ratio , we estimate a temperature for the dust of \sim  17K . For NGC 147 , we obtain an upper limit for the dust mass of 4.5 \times 10 ^ { 2 } M _ { \odot } at 160 \mu m ( assuming a temperature of \sim  20K ) , a value consistent with the previous upper limit derived using ISO observations of this galaxy . In the case of NGC 185 , we also present full 5 - 38 \mu m low-resolution ( R \sim 100 ) spectra of the main emission regions . The IRS spectra of NGC 185 show strong Polycyclic Aromatic Hydrocarbons ( PAH ) emission , deep silicate absorption features and H _ { 2 } pure rotational line ratios consistent with having the dust and molecular gas inside the dust cloud being impinged by the far-ultraviolet radiation field of a relatively young stellar population . Therefore , based on its infrared spectral properties , NGC 185 shows signatures of recent star formation ( a few \times 10 ^ { 8 } years ago ) , although its current star formation rate is quite low .