  Not every quasihereditary algebra $(A,\Phi,\unlhd)$ has an exact Borel
subalgebra. A theorem by Koenig, K\"ulshammer and Ovsienko asserts that there
always exists a quasihereditary algebra Morita equivalent to $A$ that has a
regular exact Borel subalgebra, but a characterisation of such a Morita
representative is not directly obtainable from their work. This paper gives a
criterion to decide whether a quasihereditary algebra contains a regular exact
Borel subalgebra and provides a method to compute all the representatives of
$A$ that have a regular exact Borel subalgebra. It is shown that the Cartan
matrix of a regular exact Borel subalgebra of a quasihereditary algebra
$(A,\Phi,\unlhd)$ only depends on the composition factors of the standard and
costandard $A$-modules and on the dimension of the $\operatorname{Hom}$-spaces
between standard $A$-modules. We also characterise the basic quasihereditary
algebras that admit a regular exact Borel subalgebra.
