The diversity-multiplexing trade-off (DMT) of a diamond relay channel in a quasi-static slow Rayleigh fading environment is studied, where one of the relays operates in half-duplex mode while the other one is full-duplex; hence, the term hybrid is used to refer to this channel. DMT is a known powerful tool that captures the inherent tension between the reliability and the transmission rate in a wireless communication network at high signal-to-noise ratios (SNR). In this study, we have assumed that the average SNRs of different links vary at different rates, each represented by an exponent at high SNR. These average SNR exponents are categorized into two different classes: In Class I, both of the relays are placed either in close proximity to the source or in close proximity to the destination, whereas in Class II one of the relays is closer to the source while the other one is closer to the destination. Subsequently, for Class I and for a special symmetric case of Class II, an explicit expression of the DMT of the static quantize-map-and-forward (SQMF) strategy and the optimal static schedule in the half-duplex relay, is derived. The non-symmetric case of Class II can numerically be studied using the optimization problem provided in the paper for any given parameters. As a corollary, our analysis shows that for Class I, the SQMF strategy in the proposed hybrid diamond relay channel reaches the ideal full-duplex DMT upper bound if the degrees of freedom provided by the path through the full-duplex relay is greater than or equal to that provided by the path through the half-duplex relay, or if the degrees of freedom provided by the former is less than that provided by the latter, but still large enough to support the desired multiplexing gain.