We establish upper and lower bounds on the secrecy capacity of the degraded Gaussian diamond-wiretap channel, and identify several ranges of channel parameters where these bounds coincide with useful intuitions. Furthermore, we investigate the effect of the presence of an eavesdropper on the capacity. We consider the following two scenarios: 1) common randomness is available at the source and the two relays and 2) randomness is available only at the source and there is no randomness at the relays. Our upper bounds are established by taking into account the correlation between the two relay signals and the available randomness at the encoders, which generalize the techniques recently developed for the case without secrecy constraint. For the lower bounds, we propose two types of coding schemes: 1) decode-and-forward schemes where the relays cooperatively transmit the message and the fictitious message and 2) partial decode-and-forward schemes incorporated with multicoding in which each relay sends an independent partial message and the whole or partial fictitious message using dependent codewords.