There is normally a simple choice made in the form of the covariance matrix to be used with continuous-density HMMs. Either a diagonal covariance matrix is used, with the underlying assumption that elements of the feature vector are independent, or a full or block-diagonal matrix is used, where all or some of the correlations are explicitly modeled. Unfortunately when using full or block-diagonal covariance matrices there tends to be a dramatic increase in the number of parameters per Gaussian component, limiting the number of components which may be robustly estimated. This paper introduces a new form of covariance matrix which allows a few "full" covariance matrices to be shared over many distributions, whilst each distribution maintains its own "diagonal" covariance matrix. In contrast to other schemes which have hypothesized a similar form, this technique fits within the standard maximum-likelihood criterion used for training HMMs. The new form of covariance matrix is evaluated on a large-vocabulary speech-recognition task. In initial experiments the performance of the standard system was achieved using approximately half the number of parameters. Moreover, a 10% reduction in word error rate compared to a standard system can be achieved with less than a 1% increase in the number of parameters and little increase in recognition time.