Thirteen double-mode RR Lyrae (RRd) stars, with mean magnitudes c = 18.30 ± 0.10 and <V>c = 17.80 ± 0.15, have been identified in the variable-rich Oosterhoff type I globular cluster IC 4499. The stars have surprisingly uniform properties, and are considerably different from RRd stars found in Oo II systems. The mean first-overtone period (Fig.1) is <P1>=0.357d ± 0.005d (cf. <P1>=0.40d for Oo II RRd stars), and the mean ratio of the first-overtone period to the fundamental period is <P1/P0>= 0.7443 ± 0.0002. The mean double-mode pulsation mass for the 13 stars, using the King Ia (Y=0.279, Z=0.001) mass calibration, is 0.535 ± 0.003 M⊙. Such an average mass is 0.11 M⊙ smaller (i.e. ~17% smaller) than that for RRd stars found in Oo II systems, and possibly ~0.01 M⊙smaller than the mean mass for the two RRd stars in M3 (it is important to note that the zero point of these mass determinations is uncertain by at least 15%). The metal abundances for the RRd stars, and for the system of RR Lyrae stars as a whole, are found to be consistent with <[Fe/H]>=-1.38 ± 0.20, determined from δS spectroscopy. In the Peterson diagram (Fig.2), all known RRd stars now divide (apparently by mass) into two groups (split according to Oosterhoff type). With a reddening of EB-V=0.26 ± 0.03, the cluster distance modulus is (m-M)o=16.23m ± 0.23m.