We present an analysis of ISO-SWS observations of the Herbig Ae/Be stars HD 97048 and Elias 1. Besides the well-known family of IR emission bands at 3.3, 6.2, "7.7", 8.6 and 11.2 mum these objects show strong, peculiar emission features at 3.43 and 3.53 mum. The latter two features show pronounced substructure which is very similar in the two sources. Comparison of the spectra of HD 97048 and Elias 1 with laboratory spectra of H-terminated diamond surfaces show excellent and very convincing agreement in peak position and spectral detail (Guillois et al. 1999). The position of the 3.53 mum band indicates a temperature of similar to1000 K. An analysis of the radiative energy budget makes us conclude that the diamond carrier of the 3.53 mum feature has typical sizes of 1-10 nm for HD 97048. A fit of the 3.53 mum feature with a theoretical, calculated profile indicates that the emitting diamonds in HD 97048 see a FUV flux of 5.8 x 10(-3) [W/cm(2)]. The derived diamond mass, 1.5 x 10(-10) M-circle dot, is only a tiny fraction of the total circumstellar dust mass and corresponds to only about 1 parts per billion relative to hydrogen. We discuss the origin of the diamond around these Herbig Ae/Be stars and conclude that most likely they are formed in situ. The implications for the nanodiamonds discovered in meteorites are also discussed.