  We characterise the boundedness of a Toeplitz operator on the Bergman space
with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on
the Bergman space with an L^1 symbol is completely determined by the boundary
behaviour of itss Berezin transform. This result extends known results in the
cases when the symbol is either a positive L^1 function, an L^\infty function
or a general BMO^1 function.
