The formation path to ultracompact X-ray binaries (UCXBs) with black hole (BH) accretors is still unclear. In the classical formation scenario, it is difficult to eject the massive envelope of the progenitor star of the BH via the common envelope process. Given that some neutron stars (NSs) in binary systems evidently have birth masses close to & SIM;2.0 M (& ODOT;), we explore here the possibility that BH-UCXBs may form via the accretion-induced collapse (AIC) of accreting NSs, assuming that these previously evolved in low-mass X-ray binaries to masses all the way up to the maximum limit of an NS. We demonstrate this formation path by modeling a few cases of NS-UCXBs with initial NS masses close to the maximum mass of an NS that evolve into BH-UCXBs after the NS accretes material from its He white dwarf (WD) companion. We follow the evolution of the post-AIC BH-UCXB and, based on simple arguments, we anticipate that there is about one BH-UCXB with an AIC origin and a He WD donor within the current sample of known UCXBs and that two to five such BH-UCXBs may be detected in gravitational waves by LISA. In addition, we find that the X-ray luminosity of NS-UCXBs near their orbital period minimum exceeds & SIM;10(39) erg s(-1), and thus, such systems may appear as ultraluminous X-ray sources.