We report a new type of single-electron transistor (SET) comprising two highly resistive Cr thin-film strips ($\sim1$ $\mu$m long) connecting a $1 \mu$m-long Al island to two Al outer electrodes. These resistors replace small-area oxide tunnel junctions of traditional SETs. Our transistor with a total asymptotic resistance of $110$ k$\Omega$ showed a very sharp Coulomb blockade and reproducible, deep and strictly eperiodic gate modulation in wide ranges of bias currents $I$ and gate voltages $V_g$. In the Coulomb blockade region ($|V|\leqslant$ about $0.5$ mV), we observed a strong suppression of the co-tunneling current allowing appreciable modulation curves $V(V_g)$ to be measured at currents $I$ as low as $100$ fA. The noise figure of our SET was found to be similar to that of typical Al/AlO$_x$/Al single-electron transistors, viz. $\delta Q\approx5\times 10^{-4}e/\sqrt{\mathrm{Hz}}$ at $10$ Hz.