The $\ensuremath{\Lambda}\ensuremath{\Lambda}$ hypernucleus $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ has been analyzed by use of a four-body $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ model which allows for distortion of the core by the $\ensuremath{\Lambda}$ particles. In particular, the dependence of the internal energy of the core on the rms separation of the $\ensuremath{\alpha}$ particles is required. This was obtained from three-body $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\Lambda}$ calculations for $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$. Several types of $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potentials, whose $s$-wave phase shifts had been previously obtained, were considered. Calculations for $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$ were made for a singlet $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ Yukawa potential (I) of intrinsic range $b=1.48$ F, appropriate to the exchange of two pions, and for a hard-core Yukawa potential (II) with a hard-core radius ${r}_{c}=0.42$ F and $b=2.66$ F, appropriate to a range corresponding to two pion masses for the attractive Yukawa part. Results are also given for a hard-core meson-theory potential (III) which has ${r}_{c}=0.42$ F and $b=1.48$ F. Calculations for III were made for $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{He}^{6}$, and the results were adapted to $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\mathrm{Be}^{10}$. For $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potentials which give $s$-wave phase shifts consistent with experiment, it is found that (almost independently of the details of the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ potential) the effects of core distortion account for rather more than a third of the experimental additional binding energy of 4.5\ifmmode\pm\else\textpm\fi{}0.5 MeV which is obtained after the $\ensuremath{\Lambda}$ separation energy of $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$ has been allowed for. Slightly more than half the contribution due to core distortion comes from the core energy of $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$. The remainder is due to the further distortion of the core by the second $\ensuremath{\Lambda}$, which causes approximately a 10% decrease in the rms $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ separation relative to the value for $_{\ensuremath{\Lambda}}\mathrm{Be}^{9}$. The effects of core distortion weaken the resulting $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ potential quite appreciably. For $b=1.48$ F, one obtains the scattering length ${a}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}\ensuremath{\approx}\ensuremath{-}1\ifmmode\pm\else\textpm\fi{}0.3$ F and the effective range ${r}_{0\ensuremath{\Lambda}\ensuremath{\Lambda}}\ensuremath{\approx}3.3\ifmmode\pm\else\textpm\fi{}0.6$ F, approximately independent of the shape of the $\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ potential. For II, one gets ${a}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}={{\ensuremath{-}2.3}_{\ensuremath{-}0.5}}^{+0.8}$ F and ${r}_{\ensuremath{\Lambda}\ensuremath{\Lambda}}={{4.9}_{\ensuremath{-}0.7}}^{+1.1}$ F. The well-depth parameters are 0.45\ifmmode\pm\else\textpm\fi{}0.08, 0.675\ifmmode\pm\else\textpm\fi{}0.065, and 0.77\ifmmode\pm\else\textpm\fi{}0.04 for I, II, and III, respectively. These values are about 35%, 20%, and 12%, respectively, less than the values obtained for a rigid core with a three-body ${\mathrm{Be}}^{8}\ensuremath{-}\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\Lambda}$ model. The $\ensuremath{\Sigma}\ensuremath{-}\ensuremath{\Lambda}\ensuremath{-}\ensuremath{\pi}$ coupling constant, obtained with III, is close to the value obtained from the singlet $\ensuremath{\Lambda}\ensuremath{-}N$ interaction for the same hard-core radius.