An explicit, supersymmetric solution to the low-energy heterotic string field equations representing an extended soliton with five spatial dimensions is found. This 5-brane is dual to heterotic strings in 10 dimensions in the same sense that magnetic charges are dual to electric charges in 4 dimensions. The 5-brane core is essentially a Yang-Mills instanton. The worldbrane field theory which describes the 5-brane dynamics is found to be heterotic in the sense that it has a chiral representation of supersymmetry and antichiral gauge fermions. A novel interaction between 5-branes generated by a heterotic, 2-dimensional field theory which lives on 5-brane intersection sheets, is discussed. In the context of string compactification, a similar construction is shown to lead to monopoles, strings and membranes in 4 dimensions.