Let's address your question in a canonical manner. Say that:
- the required rate of return of (private) investors is denoted by $r$.
- the interest rate or the cost of debt is denoted by $r_D$ (naturally $r_D < r$)
- the share of debt in the capital structure of the company is denoted by $X\%$
- the corporate tax rate is denoted by $T_c$
It follows that the average cost of capital (average over equity and debt), $r'$, is such that
$r' = X\% r_D (1-T_c) + (1-X\%) r$
In your case, $X\% = 100\% = 1$, then $r' = r_D(1-T_c)$.
[...] will the cost of debt be simply equal to the interest rate? If not why?
No. Because of $T_c$, the corporate tax rate being potentially non-zero.