Effective fusion of structural magnetic resonance imaging (sMRI) and functional magnetic resonance imaging (fMRI) data has the potential to boost the accuracy of infant age prediction thanks to the complementary information provided by different imaging modalities. However, functional connectivity measured by fMRI during infancy is largely immature and noisy compared to the morphological features from sMRI, thus making the sMRI and fMRI fusion for infant brain analysis extremely challenging. With the conventional multimodal fusion strategies, adding fMRI data for age prediction has a high risk of introducing more noises than useful features, which would lead to reduced accuracy than that merely using sMRI data. To address this issue, we develop a novel model termed as disentangled-multimodal adversarial autoencoder (DMM-AAE) for infant age prediction based on multimodal brain MRI. Specifically, we disentangle the latent variables of autoencoder into common and specific codes to represent the shared and complementary information among modalities, respectively. Then, cross-reconstruction requirement and common-specific distance ratio loss are designed as regularizations to ensure the effectiveness and thoroughness of the disentanglement. By arranging relatively independent autoencoders to separate the modalities and employing disentanglement under cross-reconstruction requirement to integrate them, our DMM-AAE method effectively restrains the possible interference cross modalities, while realizing effective information fusion. Taking advantage of the latent variable disentanglement, a new strategy is further proposed and embedded into DMM-AAE to address the issue of incompleteness of the multimodal neuroimages, which can also be used as an independent algorithm for missing modality imputation. By taking six types of cortical morphometric features from sMRI and brain functional connectivity from fMRI as predictors, the superiority of the proposed DMM-AAE is validated on infant age (35 to 848 days after birth) prediction using incomplete multimodal neuroimages. The mean absolute error of the prediction based on DMM-AAE reaches 37.6 days, outperforming state-of-the-art methods. Generally, our proposed DMM-AAE can serve as a promising model for prediction with multimodal data.